Download - Techniques in Broadband Interferometry · Techniques in Broadband Interferometry ... BAbout the interferometers ... I Purpose of Channel Phase Shifting

Techniques in Broadband Interferometry

A Compilation of Patents 1997-2002

David J. Erskine

Lawrence Livermore National Laboratory

December 2003 UCRL-TR-201695

This document was prepared as an account of work sponsored by an agency of the United States

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This work was performed under the auspices of the U.S. Department of Energy by University of California,Lawrence Livermore National Laboratory under Contract W-7405-Eng-48.

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Contents

Preface ix

Introduction x

1 White Light Velocity Interferometer 1-1

2 White Light Velocity Interferometer, continued 2-1

3 Noise-Pair Velocity and Range Echo-Location System 3-1

4 Multichannel Heterodyning for Wideband Interferometry,Correlation and Signal Processing 4-1

5 Single and Double Superimposing Interferometer Systems 5-1

6 Combined Dispersive/Interference Spectroscopy forProducing a Vector Spectrum 6-1

7 Examples in the Scientific Literature 7-1

Contents in detail

1 White Light Velocity Interferometer 1-1

I BACKGROUND OF THE INVENTION 1-1

A Field of the Invention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1

B Description of Related Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1

1 Superimposing interferometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1

2 Velocity sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2

3 Using sources with higher power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3

i

II SUMMARY OF THE INVENTION 1-4

III DETAILED DESCRIPTION OF THE INVENTION 1-4

A Explanation in Time-domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4B Explanation in Frequency-domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5

1 Velocity sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-52 Interferometry at multiple wavelengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-6

C Other topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-61 Transverse velocity component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-62 Relation to Optical Coherence Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . 1-6

D Complementary outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-61 Push-pull use of complementary outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-62 Practicalities using complementary outputs . . . . . . . . . . . . . . . . . . . . . . . . . . 1-7

E Example application measuring fluid flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-71 Range of allowed velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-8

F Theory regarding the superimposing condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-81 Distorting elements tolerated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-8

IV Three modes of measuring fringes 1-9

A Single detecting channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-9B Multiple detecting channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-9

1 Wavelength distributed system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-92 Delay distributed system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-103 Narrow-band illumination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-104 Chirped illumination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-11

V Interferometer design classes 1-12

A Michelson vs. Fabry-Perot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-12B Superimposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-13

1 Tolerance for smear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-13C Fabry-Perot Interferometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-14D Superimposing delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-14

1 Etalon delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-152 Delays using real imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-153 Lens delays using virtual imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-154 Minimizing chromatic aberrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-165 Creating negative and positive delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-166 Computing the net delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-177 Use with optical waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-178 Why put image at mirror plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-189 Use with sound or microwaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1810 One-dimensional configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1811 For matter waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-19

ii

VI Example apparatus using retro-reflected light 1-19

Acknowledgments 1-19

VII CLAIMS 1-20

2 White Light Velocity Interferometer, continued 2-1

I CLAIMS 2-1

3 Noise-Pair Velocity and Range Echo-Location System 3-1

I Introduction 3-1

A Field of the Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1

B Description of the Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1



A Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-3

B About the interferometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-4

C Measuring a Doppler Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-4

D Velocity from Phase Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-5

E Equivalence to an Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-5

F Targets Separated from Clutter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-6

G Clutter Cancellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-8

H Plotting the Autocorrelation vs Wavelength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-9

I Sensing Target Albedo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-9

J Pulse Repetition and Echolocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-9

K Explanation in Frequency Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-11

1 Undesired Kind of Illumination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-12

2 Comparison to Matched Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-12

L Expected Practice for Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-13

1 Moving Reference Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-13

M Several Embodiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-13

1 Ultrasound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-14

2 General Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-14

N Simultaneous Multiple Banded Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-14

1 Means for Achieving Long Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-14

O Optical Embodiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-17

iii


IV CLAIMS 3-21

4 Multichannel Heterodyning for Wideband Interferometry,Correlation and Signal Processing 4-1

I Introduction 4-1





A Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-6

B Down-mixer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-6

C Vector signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-8

D Channel signal processors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-9

E Up-mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-10

F Summation over channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-11

G Phase Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-11

H Temporal and Spectral Dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-11

I Purpose of Channel Phase Shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-12

J Rotation of the IF Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-13

K Channel Interferometer with Vector Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-13

L Channel Delay Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-15

M Channel Recirculating Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-16

N Channel Arbitrary Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-16

O Channel Autocorrelator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-18

P Sinusoidal Weighting After Time-averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-19

Q Channel Correlator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-22

R Channel Recorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-24

S Channel Waveform Synthesizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-24

T Combination Signal Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-24

U Calculating Desired Channel Phase Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-25

1 Plotting phase versus frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-25

2 An Initially Phase-nonlinear Device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-25

3 Bringing it into Phase-linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-25

V Creating Small Changes in Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-25

iv

IV Master Phase Equation 4-25

A When Using a Pair of Devices Sharing fLO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-26

B Numerical Example of Choosing Phase Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-27

C Practical Advantages of Sharing Reference Frequencies . . . . . . . . . . . . . . . . . . . . . . . . 4-27

D Tolerance for Reference Frequency Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-27

E An Autocorrelator Sharing Reference Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-27

F A Correlator Sharing Reference Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-28

G Synchronizing Widely Spaced Correlator Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-28

H Correlator for Optical Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-28


V CLAIMS 4-29

5 Single and Double Superimposing Interferometer Systems 5-1

I BACKGROUND OF THE INVENTION 5-1

A Field of the Invention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1


1 Superimposing Interferometers and Uncollimated Light . . . . . . . . . . . . . . . . . . . 5-1

2 Coherent Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1

3 Angle Dependence of Classic Michelson and Fabry-Perot Interferometers . . . . . . . . . . 5-2

4 Superposition Creates Angle Independence . . . . . . . . . . . . . . . . . . . . . . . . . . 5-4

5 Superimposing Fabry-Perot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-4

6 Virtual Imaging by an Etalon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-5

7 Double Interferometers for White Light Velocimetry . . . . . . . . . . . . . . . . . . . . . 5-5

8 Broad Bandwidth Illumination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-6

9 Velocimetry with a Double Fabry-Perot . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-6

10 Beam Shortening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-7


v


A Michelson-class interferometers using real imaging . . . . . . . . . . . . . . . . . . . . . . . . . . 5-9

B Manifold of element positions for a relay delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-11

C Applications for an adjustable delay superimposing interferometer . . . . . . . . . . . . . . . . . 5-12

D Multi-stage relay delay and waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-14

E Delaying mirrors using virtual imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-14

F Wedge and inclined delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-16

G Achromatic inclined delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-17

H Achromatic Etalon Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-18

I Differential interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-19

J Increasing fringe visibility by A.M. placement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-19

K Recirculating interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-20

L Interferometers made without delaying mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-21

M Zero delay applications of superimposing interferometers . . . . . . . . . . . . . . . . . . . . . . . 5-23

N Electrical circuit equivalent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-24

O Two-delay interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-25

P Delaying a beam using a delaying mirror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-25

Q Superimposing interferometer in series with spectrometer . . . . . . . . . . . . . . . . . . . . . . 5-26

R Double superimposing interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-28

1 Optical communication with a double superimposing interferometer . . . . . . . . . . . . 5-29

2 Matching delay and dispersion in a double superimposing interferometer . . . . . . . . . . 5-29

3 Matching dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-31

4 Constant-delay mirror motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-32

5 Matching procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-32

6 Retro-reflecting overlap adjustment subprocedure . . . . . . . . . . . . . . . . . . . . . . . 5-32

7 Stationary fringe overlap adjustment subprocedure . . . . . . . . . . . . . . . . . . . . . . 5-33

S Retro-reflecting interferometer as double interferometer . . . . . . . . . . . . . . . . . . . . . . . 5-33

T Methods of Discriminating Beams in a Retroreflecting Interferometer . . . . . . . . . . . . . . . . 5-33

1 Discrimination by beam angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-33

2 Discrimination by polarization and image offset . . . . . . . . . . . . . . . . . . . . . . . . 5-34

U Birefringent wedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-34

V Other offset target image methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-35

W Kinds of waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-36


IV CLAIMS 5-37

6 Combined Dispersive/Interference Spectroscopy forProducing a Vector Spectrum 6-1

vi

I Introduction 6-1


B Description of Related Art: Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1

1 Doppler Velocimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1

2 Drawbacks of Purely Dispersive Spectrosgraphs . . . . . . . . . . . . . . . . . . . . . . . . 6-2

3 Drawbacks of Purely Interferometric Spectrosgraphs . . . . . . . . . . . . . . . . . . . . . 6-3

4 Prior Dispersive Interferometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-3

5 Prior Non-dispersed Interferometer for Metrology . . . . . . . . . . . . . . . . . . . . . . . 6-3

6 Prior use of Interferometry for Angle Measurement . . . . . . . . . . . . . . . . . . . . . . 6-3


A Combine Dispersion with Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-4

B Spatial or Temporal Phase Stepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-4

C A Heterodyning Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-4

D Vector Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-4

E Use for Doppler Velocimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-4

F Use for Metrology of Secondary Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-5

G Use for Measuring Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-5

H Use for Mapping a Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-5


A An Embodiment For Doppler Velocimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-6

1 Spectral Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-6

2 Optics Preparing Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-7

B The Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-7

1 Fringes in Frequency and/or Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-8

2 Fixed Delay Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-8

3 Wide-angle Ability from Superimposing Condition . . . . . . . . . . . . . . . . . . . . . . 6-8

4 Finding the Delay from the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-8

5 Two Complementary Outputs Available . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-9

6 Means for Phase-stepping the Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-9

7 Temporarily Preventing Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-9

8 Finite vs Infinitely Tall Spatial Fringes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-9

9 Best Size of Spatial Fringe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-10

10 Case of Very Many Spatial Fringes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-10

C The Disperser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-10

1 Combating Astigmatism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-10

2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-11

3 Use of a 2nd Interferometer for Fiducials . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-11

vii

IV Theory of Operation 6-11

A Moire or Heterodyning Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-11

B Interferometer Spectral Comb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-11

C Benefit of Randomness of Systematic Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-12

D Relation Between Velocity and Moire Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-13

V Data Taking Procedure 6-13

VI Making the Whirl from the Fringing Spectrum 6-13

A Finite Fringe Period Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-13

B Infinite Fringe Period Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-13

C Phase Stepping to Reduce Fixed-pattern Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-14

D Finding Whirl Rotation for Determining Doppler Shift . . . . . . . . . . . . . . . . . . . . . . . . 6-14

1 Dot Product Between Two Whirls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-14

2 A More Exact Dot-product Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-15

3 High Precision Verified by Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-15

E Metrology Based on Dilation of an Interferometer Cavity . . . . . . . . . . . . . . . . . . . . . . 6-15

VII Measurement of Angles Using a Long-baseline Interferometer 6-17

1 Iodine Cell Imprints a Reference Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 6-17

2 Relation Between Target Angle and Whirl Rotation . . . . . . . . . . . . . . . . . . . . . 6-18

3 Works with Non-zero Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-18

4 Use of Reference Stars to Take Angular Difference Independent of Pathlength Drifts . . . 6-18

5 Distinction from Prior Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-19

VIII Spectral Mapping 6-19

A Stepped Mirror and Etalon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-19

B Explanation of Behavior through the Interferogram . . . . . . . . . . . . . . . . . . . . . . . . . . 6-21

C Data Analysis for Reconstructing the Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-21


IX CLAIMS 6-22

7 Examples in the Scientific Literature 7-1

I White Light Velocity Interferometry 7-1

II Dispersive Interferometry (Astronomical Velocimetry) 7-2

III Dispersive Interferometry (Angular Measurement) 7-3

IV Dispersive Interferometry (High Resolution Spectroscopy) 7-4

viii

Preface

This is a compilation of my patents issued from 1997 to 2002, generally describing interferometertechniques that modify the coherence properties of broad-bandwidth light and other waves, withapplications to Doppler velocimetry, rangefinding, imaging and spectroscopy.

Patents are tedious to read in their original form. In effort to improve their readability Ihave embedded the Figures throughout the manuscript, put the Figure captions underneaththe Figures, and added section headings. Otherwise I have resisted the temptation to modifythe words, though I found many places which could use healthy editing. There may be minordifferences with the official versions issued by the US Patent and Trademark Office, particularlyin the claims sections.

David J. Erskine

Lawrence Livermore National LaboratoryDecember, 2003

[email protected]

ix

Introduction

In my shock physics work I measured the velocities of targets impacted by flyer plates byilluminating them with laser light and analyzing the reflected light with an interferometer.Small wavelength changes caused by the target motion (Doppler effect) were converted intofringe shifts by the interferometer. Lasers having long coherence lengths were required for theillumination.

While lasers are certainly bright sources, and their collimated beams are convenient to workwith, they are expensive. Particularly if one needs to illuminate a wide surface area, then largeamounts of power are needed. Orders of magnitude more power per dollar can be obtainedfrom a simple flashlamp, or for that matter, a 50 cent light bulb. Yet these inexpensive sourcescannot practically be used for Doppler velocimetry because their coherence length is extremelyshort, i.e. their bandwidth is much too wide. Hence the motivation for patents 1 & 2 isa method (White Light Velocimetry) for allowing use of these powerful but incoherentlamps for interferometry. The coherence of the illumination is modified by passing it througha preparatory interferometer.

This method developed for light would also work for other wavelengths and kinds of waves–such as microwaves for radar and sound for sonar and medical imaging. In these applicationsthe ability to measure distance, not only velocity, was important. In the prior art, measuringboth simultaneously require two kinds of sources, velocity using a long coherence length sourceand range using very short coherence length source. That is, they could not be measuredsimultaneously with the same apparatus. Hence patent 3 explored methods of measuringrange and Doppler velocity simultaneously with a single short coherence length source. Thename “Noise-Pair Radar” refers to the use of incoherent illumination that is sent through apreparatory interferometer to generate an echo. Both the original signal and echo illuminatethe target.

The above method requires construction of delay lines (i.e. interferometers) for microwavefrequencies that are very long compared to the wavelength. To accomplish this with simplecables could be impractical, due large weight and cost. Patent 4 (Multichannel Heterodyn-ing) solves this problem by subdividing the wide bandwidth of the signal into smaller chunksand lowering their frequencies through a heterodyning effect to a range where inexpensiveelectronics can delay them.

The White Light Velocimetry method also requires interferometers and delays lines thatare long compared to the wavelength, but in the optical regime. Patent 5 (SuperimposingInterferometers) describes methods of creating these interferometers. These work with theuncollimated beams characteristic of ordinary lamps, rather than the convenient pencil beamsof lasers. The patent also describes several applications which can be done by combinations ofinterferometers, such as encrypted communications (page 5-29).

Excited about the news that astronomers could detect exoplanets by measuring Dopplershifts in white starlight, and having already measured similar magnitude velocities in whitelight interferometrically, I explored ways of helping their effort. Being an interferometer special-ist, my approach was naturally biased toward interferometry rather than the purely dispersivespectrographs of the prior art. Since I had already explored the combination of dispersion with

x

interferometry in patent 1 (see Fig. 5, page 1-10), that became my approach. The difference wasthat for astronomical velocimetry the target was not actively illuminated but emitted its ownlight having identifiable spectral features. Hence the preparatory interferometer of Fig. 5 waseliminated. This led to patent 6, called a variety of names including Fringing Spectroscopy, Ex-ternally Dispersed Interferometry, Spectral Interferometry, Dispersive/Interference VectorSpectroscopy.

Developing this technique led to ancillary applications. The ability to precisely measurea white light fringe shift– needed to measure small Doppler velocities– can also measure theangular position of a star, if the interferometer is modified to have two inputs separated by abaseline (Spectral Astrometry, page 6-17). The significant spectral dispersion allows a newability not shared by the prior art– the ability to measure two or more stars simultaneously onthe same detecting CCD chip. This allows for the first time differential angular measurementsthat are dramatically insensitive to drifts in the optical path length of the baseline. This solvesa significant engineering problem of the prior art.

Another ancillary application is enabling the spectrograph to detect higher spectral resolu-tion features than when used without the interferometer (spectral mapping or high resolutionspectroscopy, page 6-19, also called Resolution Boosting). The moire patterns created inthe spectrum are numerically processed to recover high spectral details otherwise blurred bythe spectrograph. Hence the interferometer can be added to an existing spectrograph systemlike “eyeglasses” to improve the system’s performance at low cost.

xi

IL-9745WLVpat10c.tex 3/05

Part 1

White Light VelocityInterferometer

US Patent 5,642,194Issued June 24, 1997

David J. Erskine∗Lawrence Livermore Nat. Lab.

The invention is a technique that allows the use of unlimited bandwidth and extended illuminationsources to measure small Doppler shifts of targets external to the apparatus. Monochromatic andpoint-like sources can also be used, thus creating complete flexibility for the illumination source.Although denoted white light velocimetry, the principle can be applied to any wave phenomenon,such as sound, microwaves and light. For the first time, powerful, compact or inexpensive broadbandand extended sources can be used for velocity interferometry, including flash and arc lamps, lightfrom detonations, pulsed lasers, chirped frequency lasers, and lasers operating simultaneously inseveral wavelengths. Superimposing interferometers are created which produce delays independent ofray angle for rays leaving a given pixel of an object, which can be imaged through the interferometer.

I. BACKGROUND OF THE INVENTION

A. Field of the Invention

The present invention relates to velocimetry, and morespecifically, it relates to the use of a velocity interferom-eter capable of using unlimited bandwidth illumination.

B. Description of Related Art

Measuring the velocity of an object remotely throughthe Doppler shift of reflected wave radiation (electromag-netic waves, sound) is an important diagnostic tool ina variety of fields in science and engineering. In shockphysics, velocity interferometry is an important opti-cal technique measuring velocities of impacted targets,typically moving 1-10 km/s. In law enforcement andmeteorology, other kinds of Doppler velocimeters usingmicrowaves measure velocities of automobiles and rain-drops. And in medicine using ultrasound, Doppler in-struments detect the motion of blood in vivo. All thesevelocimeters require coherent quasi-monochromatic illu-mination, which is generally more expensive and lowerpower than broadband incoherent sources, for any kindof wave radiation. For example using light, for the singleinterferometer velocimeter, the illumination was previ-ously restricted to lasers operating in a single longitu-dinal frequency mode, which reduces their power. Forthe double interferometer velocimeter (U. S. Patent No.4,915,499), the illumination was restricted to less than

∗[email protected]

4 nm for typical laser illumination, and would be evenless for an extended source. In comparison, visible light(red to blue) has a bandwidth of approximately 200 nm.(The limited bandwidth was calculated by the authorsof that patent in an article “Multiple-line laser Dopplervelocimetry” by S. Gidon and G. Behar, Applied OpticsVol. 27, p2315-2319, 1988, on page 2317, below Equa-tion 11, and assumed a typical optical arrangement.)

The invention and the prior art velocimetry discussedhere is characterized by a single illuminating beam strik-ing the target. This is distinguished from another widelyused velocimetry technique using two intersecting laserbeams in a transparent fluid, in which scattering particlescarried by the fluid pass through a standing wave grat-ing created by the intersecting beams. That techniqueis commonly called LDV, for laser Doppler velocimetry.(The title is not helpful in distinguishing the differentkinds of prior art, since they all are based on Dopplershifts and almost all on laser illumination.)

1. Superimposing interferometers

The prior art of velocity interferometry (single beamstriking the target) can be classified into kinds using asingle interferometer, or two interferometers before andafter the target. The former is much more common.For use with light, single interferometer velocimeters aredistinguished by the kind of interferometer. Those us-ing monochromatic superimposing Michelson interferom-eters are called VISARs (velocity interferometer systemfor any reflector), and those using non-superimposingFabry-Perot interferometers are called Fabry-Perot ve-locimeters. The latter are non-superimposing becauseof their design, (and their method of use depends on

1-2

14

1012 16

1

13

13

25

23

4

3

Figure 1A

s (1-2v/c)

+ d

18 20

22 24

Figure 1B

Constructive or destructiveinterference

Figure 1A illustrates the white light velocimeter concept of the present invention. Figure 1B shows the waveform after thedetecting interferometer when the source is a single wave packet.

the non-superimposing nature through the creation ofoff-axis fringes which shift in angle with velocity.) Thedefinition of superimposing will be defined later. A dou-ble interferometer using two non-superimposing Fabry-Perot interferometers is described in U. S. Patent No.4,915,499.

The prior art either uses single interferometers, or dou-ble interferometers lacking in the proper superimposingquality. The prior art did not recognize that in orderto use unlimited bandwidth illumination from an ex-tended source, two achromatic superimposing interferom-eters are needed, before and after the target. (The twocan be optionally conveniently accomplished with oneset of interferometer optics by retro-reflecting the lightfrom the target.) Furthermore, the superimposing de-signs presented in the present invention are achromatic,whereas the prior art of single superimposing interfer-ometers (VISARs) suffered chromatic aberrations whichlimit their use to monochromatic light.

The distinction between the superimposing quality ofthe present invention and the non-superimposing qual-ity of the prior art is important to understand, and iscrucial in creating many of the beneficial capabilitiesof this invention. The non-superimposing velocimetercannot imprint a constant delay on all the light froma non-directional extended source, such as an incandes-cent lamp, due to the slight dependence of the delay onray angle. Thus, it cannot form an imaging velocimeterhaving a constant imprinted delay across the image. Forthe non-superimposing interferometer, there is a tradeoffbetween maximum bandwidth and degree of parallelism

of the source rays. This in turn prevents many desirablecapabilities that come from a wide bandwidth, such assimultaneous multiple velocity detection, unambiguousvelocity determination, independence from target and il-lumination color, and lack of speckle.

2. Velocity sensitivity

The velocity sensitivity of all the velocity interferome-ters discussed, single or dual interferometer, superimpos-ing or non-superimposing, broad or narrowband, all havethe same fundamental relationship between fringe phaseshift for a specific color, and target velocity. This willbe described now. The additional advantages that comewith broadband illumination will be discussed later.

The single interferometer velocimeter consists of amonochromatic source illuminating the target, and aninterferometer analyzing the reflected radiation. If theinterferometer is a Michelson design, it has a fixed delay τbetween its two arms. In the case of a Fabry-Perot inter-ferometer, τ describes the delay between multiple outputpulses produced from a single input pulse, if the illumina-tion was considered hypothetically to be a stream of inde-pendent pulses. The nonzero value of the delay convertssmall Doppler shifts of the reflected light spectrum intofringe shifts in the interferometer output. Fringes denotethe fluctuations in time-averaged output intensity thatare not due to fluctuations in the input intensity. Thesecan vary sinusoidally, or at least periodically, with targetvelocity v, wavelength λ and delay τ . There is a propor-

1-3

tionality η between the target velocity and fringe phaseshift for a specific wavelength of illumination describedby

η =λ

2τ(1)

and λ is the average wavelength of light. For example,to measure highway velocities in green light, η should beof the order 10 m/s, which requires cτ ≈ 8 m, where cis the speed of light. (Delays are conveniently describedeither by duration τ , or the distance light travels in thattime cτ .) Equation 1 neglects dispersion in the glassoptics inside the interferometer, assumes v/c 1 andthat light reflects normally off the target.

If light does not reflect normally off the target, thevelocity component being measured can always be de-fined as half the rate of change of the effective roundtrip path length from the source to the target to the de-tecting optics. The effective path length is the integralof the refractive index along the path. This definitionaccounts for any arrangement of source, target, targetmotion vector, detecting optics and changes in refractiveindex of the medium or target, and applies to all Dopplervelocimeters.

The velocimeter can measure target displacement orvelocity, depending on the time scale of the recordingsystem. If the detector response time is very fast, fasterthan the interferometer delays, but still slower than asystem coherence time Λsys/c, then the fringe shift ∆φ isrelated to a displacement of the target during the intervalτ according to

∆φ =2d

λ(2)

where d is component of the target motion toward thesource and detecting interferometers. This is not the typ-ical situation. Typically the detector or recording equip-ment response time is slower than the delay time τ . Inthis case, the fringe shift is related to velocity accordingto

∆φ =2vτ

λ(3)

The velocity component of the target toward the ve-locimeter is obtained from the fringe phase shift ∆φ by

v = ∆φ η =λ

2τ∆φ (4)

3. Using sources with higher power

In previous velocimeters, the coherence length (Λ) ofthe illumination must be as large as cτ in order to pro-duce fringes with significant visibility. This severely re-stricted the kind of light source which could be used.The Λ of white light (∼1.5 µm) was insufficient. Pre-viously, lasers were the only light sources used because

their coherence length could be made sufficiently longwhen operated in a single frequency mode. However, inthis mode the output power is low.

The low output power restricted the applications ofoptical velocimetry. Typical laboratory measurements inshock physics were limited to measurement of velocityat a single point on the target. These experiments, andmany other laboratory and industrial applications can begreatly improved if sample velocity could be measuredsimultaneously at more than one point, such as over aline or an area. For example, to measure the velocityfield over an area at a single moment in a wind tunnel,or to measure the complex motion of an exploding non-symmetrical object.

However, measuring velocity over a line or an area in asnap-shot requires orders of magnitude more power thanwhat can be provided by single mode lasers. Amplifierscan be used to boost laser power, however, these are ex-pensive and bulky. Furthermore, velocimetry of a remoteobject through a telescope in the field demands orders ofmagnitude more power, since reflected light intensity isweak. Previously, the velocity can be measured over anarea by scanning a single point measuring beam. How-ever, this is not appropriate for measuring complex dy-namic events such as turbulence, which grow on a timescale faster than the scanning can be completed.

In contrast, the invention we describe can measure ve-locities simultaneously over an area, taking advantageof the velocimeter’s imaging capability, and the higherpower available from inexpensive and compact broad-band extended sources. These sources are not suitablefor a conventional velocity interferometer due to theirnon-parallel rays and broad bandwidth. Even the priorart double Fabry-Perot velocimeter can only use but afraction of the available power because of the trade-offbetween maximum bandwidth and degree of parallelism.In contrast, the superimposing interferometers describedin this application create a fringe phase which is constantover a range of ray angles.

For the purpose of illuminating a target over a widearea, the directional quality of a laser beam is not acomparative advantage, since the laser light must bedispersed to cover the area, and the target will usu-ally randomly scatter the reflected light in all directions.For example, a $50,000 argon laser operating in single-frequency mode only produces approximately 1 watt ofpower. In contrast, a flashlamp costing a $1000 dollarscan create perhaps a 100,000 watts. If the superimposinginterferometers are achromatic and have a large numer-ical aperture capability, then a large fraction of the rayangles and colors of the flashlamp can be utilized forwhite light velocimetry. Thus, for illuminating an area,more illuminating power can be obtained from the flash-lamp than the laser, and for less expense.

1-4

II. SUMMARY OF THE INVENTION

It is an object of the present invention to measure thevelocity, or displacement during a short interval, of ob-jects using light, microwaves or sound from sources whichmay be broadbanded and may be extended in emittingarea. This requires two achromatic superimposing inter-ferometers in series, with the target interposed, topolog-ically.

It is another object of the invention to delay the lightrays of one interferometer arm in time, yet superimposethem in path, to create a superimposing delay and hencea superimposing interferometer. It is an object to makethese delays achromatic, that is, satisfy the superimpos-ing conditions for a wide range of wavelengths and foruncollimated rays.

It is another object of the invention to optionally mea-sure velocity at more than just a single point on the targetsurface, by forming a line or a 2-dimensional image of thetarget at the detector which manifest fringes from whichthe velocity is deduced.

It is another object of the invention to optionallyuse illumination whose wavelength changes with time(chirped), in conjunction with a spectrally or delay-organized detector, so as to record velocity and reflec-tivity of objects versus time in analogy to an electronicstreak camera.

The first object is achieved only through the simulta-neous use of two conditions 1) two delay-matched inter-ferometers in series with the target interposed; and 2)interferometers which are achromatically superimposing.When the interferometer delays match within a systemcoherence time, partial fringes are produced which shiftwith target velocity via the Doppler effect. The inventionis a generic method for velocity interferometry using anysource, broad or narrowbanded, pointlike or extended.The principle is applicable to any wave phenomenon forwhich superimposing interferometers can be constructed,such as light, sound, and microwaves. Powerful flash-lamps, which are extended and broadbanded, can be usedfor velocity interferometry where they were prohibitedbefore. Impulsive sound created by pulses of laser lightcommunicated to a patients body through optical fiberscan now be used for Doppler ultrasound imaging, wherepreviously it was prohibited due to incoherence.

The fourth object will form a device capable of mea-suring velocities and reflectivities of objects down to 2ps time resolution when used with chirped illumination.This embodiment is called a chirped pulse velocimeter.“Chirped” light is defined herein as light which instan-taneously has an approximately pure wavelength thatvaries with time. It forms a kind of optical streaking cam-era, similar to an electronic streak camera, whose timeresolution is not affected by the motion of electrons, butby the wave properties of light.

The superimposing requirement for the interferometersis mandatory in the case of waves which travel in media inmore than one dimension. If the waves are expressed as

a 1-dimensional quantity, such as the voltage in an elec-trical cable leading from an antenna or a transducer, thesuperimposing requirement is moot, since there is onlyone ray path, not a manifold of paths. In that case, theinterferometers can use a simple time delay, implementedby electronics or mathematically implemented throughsoftware.

The present invention is distinguished from the priorart by the achromatic superimposing quality of the inter-ferometers, which can be constructed to operate on wavesof a general nature, not necessarily light. In this appli-cation, the term “white” denotes the broadband qualityof waves of any kind, not restricted to light, such as mi-crowaves and sound.

III. DETAILED DESCRIPTION OF THEINVENTION

A. Explanation in Time-domain

Figure 1A illustrates the white light velocimeter con-cept of the present invention. A coherent echo is im-printed on the source light 10 by a superimposing in-terferometer (denoted the source interferometer 12) hav-ing a delay τs. The reflected light from the target 14 isobserved through a second superimposing interferometer(denoted the detecting interferometer 16) having delayτd. This forms complementary outputs 3, 4. Figure 1Ais a topological diagram only, the detailed positions ofoptics internal to the interferometers is not meant to bedepicted. The time averaged output intensity is observedby a detector 13 or camera, which could be a single chan-nel or multi-channel device. Partial fringes result whenτs ≈ τd.

Suppose white light is modeled by a series of indepen-dent wave packets; then we can follow the result of a sin-gle wave packet from the source. The waveform leavingthe detecting is shown in Figure 1B. The source inter-ferometer 12 splits the single packet into two identicalpackets (18, 20). The detecting interferometer does thesame to create four of the two (22, 24). Two of thosepackets (20 , 22) will overlap and interfere if τs ≈ τd,within a coherence length, contributing fringes to the netdetected intensity 23. Target 14 motion during the inter-val τs will change apparent τs, so that it appears to beτs(1 − 2v/c) to the detecting interferometer. This is theDoppler effect, which contracts or dilates the waveform,or equivalently dilates or contracts the spectrum, causinga fringe shift in the detecting interferometer output in-tensity 23. The velocity/displacement is approximatelygiven by the fringe phase shift according to Eq. 4 or Eq. 2,where λ is the average wavelength of the light reachingthe detector or specific detecting channel.

The other packets (18, 24) add incoherently and con-tribute nonfringing components. Thus the visibility ofthe fringes are always partial (although use of a high fi-nesse Fabry-Perot as the source interferometer 12 can

1-5

0

T( )

Frequency ( )

29 30

27

31

100%

Fig. 2

Figure 2 compares T (ω) for source and detecting interferom-eters.

arbitrarily increase fringe visibility at the expense ofthroughput). The fringe visibility can be made 100% bysimultaneously recording both complementary detectinginterferometer outputs (23, 25) and numerically or elec-tronically subtracting them. This also eliminates extra-neous light, which is defined light that has not passedthrough the source interferometer.

B. Explanation in Frequency-domain

An equivalent explanation of the basis of operation isto consider each interferometer to be a filter which has aperiodic power transmission spectrum T (ω), also calleda “comb” filter. This explanation is valid when the de-tector time response is slower than 2τ , which is the usualcase. The angular frequency ω is related to the ordi-nary frequency and wavelength through ω = 2πf = c/λ.Figure 2 compares T (ω) for source and detecting inter-ferometers. For each comb filter 29, 30, the periodicityin frequency 27, 31 is inversely proportional to the timedelay. If τs = τd and the target is stationary, then trans-mission peaks of the source comb filter overlap peaks ofthe detecting comb filter. This produces a “bright” fringein the output. If the target moves, the source spectrumis changed (multiplied by 1 + 2v/c) by the Doppler ef-fect, so that the peaks of the source filter may no longercorrespond to the peaks of the detecting comb filter, asdepicted in Figure 2. This produces a “dark” fringe inoutput intensity of the light passing through both in-terferometers, integrated over all frequencies. (This issimilar to the creation of a Moire pattern from two finelyspaced grids.) Depending on the illumination bandwidth,the integrated overlap of source and detecting comb fil-ter peaks goes in and out of phase as the target velocityincreases, producing a sequence of light or dark fringes.

The description of the time-averaged detecting inter-ferometer output intensity 〈I〉 for a target velocity v isgiven by the overlap integral over angular frequency ω.

〈I〉 ∝∫ ∞

0

dω S(ω/D2) Ts(ω/D2) Rt(ω/D1) Td(ω)P (ω)

(5)

where

D2 = (1 + 2v/c) (6)

and

D1 = (1 + v/c) (7)

are Doppler multipliers, valid when v c. S(ω) andP (ω) are the power spectra of the illumination and de-tector sensitivities, respectively. The functions Ts(ω) andTd(ω) are the power transmission spectra of the sourceand detecting interferometers respectively. These are cal-culated by the magnitude squared of the Fourier trans-form of the impulse response in the electric field.

If the detector time response is much shorter than thecoherence time for the detecting channel, then the Equa-tions stated here may not apply. If the detector timeresponse is greater than or equal to the coherence time,then the Equations stated here apply, provided the veloc-ity is constant during the observation time. If the targetis accelerating, the fringe will shift, and if the fringe shiftsmore than 90 during the detector response time, the netfringe will wash out.

An idealized dispersionless Michelson interferometer (a50% reflective beamsplitter is assumed for this discus-sion) has

T (ω) = 12 (1 + cos ωτ) (8)

A superimposing Fabry-Perot interferometer has

T (ω) = [1 + α sin2(ωτ/2)]−1 (9)

mirror reflectivity R by α = 4R/(1 − R)2. Here we as-sumed both cavity mirrors have the same reflectivity.However, this is not required in general.

For the common case of Michelson source and detectinginterferometers, fringes in the output intensity describedby Equation 5 will occur when τs ≈ τd, within a coher-ence time Λ/c. That is, these fringes will be sinusoidalunder an envelope having a width Λ/c. At the center ofthe envelope, the intensity will vary approximately as

〈I〉 ∝ cos[2π

λ(cτs − cτd) −

2π

λ(cτs) (2v/c) + φ0

](10)

where φ0 is some phase constant. For Michelson interfer-ometers with 50% transmitting/reflecting beamsplitters,the amplitude of the fringes at the peak of the envelopeis a visibility of 50%, and less than this on the wings ofthe envelope.

1. Velocity sensitivity

The velocity can be calculated from the shift of anypart of the fringe pattern, either the envelope or thephase of the fringes underneath the envelope. If thefringe phase is used, then the velocity/displacement isfound from Eq. 4 or Eq. 2. If the shift of the envelopeis used, then the shift is first expressed in terms of delaylength which could be denoted c∆t. Then the displace-ment is found through d = c∆t/2 and the velocity isfound through v = c∆t/2τ .

1-6

2. Interferometry at multiple wavelengths

Note that the fringes of Equation 10 vary sinusoidallyversus velocity, frequency (c/λ), and difference in inter-ferometer delay (τs-τd). Thus the fringe behavior canbe measured and analyzed versus frequency, or delay, orcombinations of both. If the fringe behavior is decom-posed into components organized by wavelength, eachcomponent will have an individual phase shift which canbe measured and a velocity value computed from eachthrough Eq. 4 using the specific wavelength associatedwith each component. These velocity values should allagree if the correct integer order is guessed when measur-ing the phase. Thus, the use of several wavelength com-ponents provides a self-consistency check on the analysisthat resolves integer fringe skip ambiguities.

C. Other topics

1. Transverse velocity component

Assuming that the light from illuminating source10 reflects nearly normally from the target 14, thenthe term velocity means the component toward thesource/detector. However, the source 10 can illuminatelaterally, or approximately on the opposite side of thetarget 14 so that a transverse velocity component is mea-sured. This could be advantageous in fluid flow measure-ments. The illumination could pass through the target,such as if the target were a region of plasma whose refrac-tive index was being probed. In all cases, the term “2v”in the Equations represents the rate of increase of theround trip effective path length source-target-detector,where the effective path length is the refractive index ofthe medium integrated along the physical path.

2. Relation to Optical Coherence Tomography

In the sense that the two inner pulses (20, 22) of Fig-ure 1B are being compared by time of arrival, white lightvelocimetry is related to optical coherence tomography(OCT), which uses incoherent light to measure differ-ences between two interferometer mirror displacements.The distinction is that in OCT, the target plays the roleof one of the interferometer mirrors, that is, the targetis internal to an interferometer. In white light velocime-try, the target is external to any interferometer. This isa great advantage for uncooperative targets, since strictalignment is not required, as in OCT.

D. Complementary outputs

In Figure 1A, the box-like constructions for source in-terferometer 12 and detecting interferometer 16 are icons

representing generic amplitude-splitting interferometerswhich satisfy the superposition condition. The com-plementary outputs are not always drawn, but are im-plied. Their use is not mandatory, but is usually help-ful. Michelson, Mach-Zehnder, Sagnac, or special Fabry-Perot design interferometers can be modified to satisfythe superposing condition and therefore be usable in thisvelocimeter (the ordinary Fabry-Perot design will notwork because it is not superimposing). The source in-terferometer 12 and detecting interferometer 16 need notbe the same design. For example, the source and targetmay have different numerical aperture requirements, sothat optimal designs are different. Also, since the com-plementary source interferometer output is discarded inthe simplest configuration, the source interferometer canhave a more streamlined designed, such as an inline su-perimposing Fabry-Perot.

On the other hand, if it is desired, the role of both in-terferometers can be performed by the same optics by ob-serving the light retro-reflecting from the target 14 backtoward the source. This is accomplished simply by iden-tifying in the single interferometer the output beam com-ing from the target. This automatically satisfies τs ≈ τd

and greatly simplifies alignment. However, separate in-terferometers will be desired for the case of illuminationthat is very intense relative to reflected light, to avoidthe glare from shared optics.

1. Push-pull use of complementary outputs

It is recommended, but not necessary, that both com-plementary outputs 3 & 4 of the detecting interferometerbe recorded. By numerically or electronically subtractingtheir signals, any background light that has not passedthrough source interferometer will be eliminated from thesignal. This is called “push-pull” processing of the sig-nals. Secondly, the determination of fringe phase willbe much more accurate for fluctuating target reflectivity,since the difference signal will be known to be centeredabout zero. Thirdly, when the velocimeter is used in asingle-channel velocity discriminator mode, only the ob-jects having the required velocity will produce a nonzeroimage in the difference signal. Fourthly, if the comple-mentary signals are added instead of subtracted, thenthe non-fringing (i.e. ordinary) image will be obtainedwithout speckle. This is useful for the presentation ofthe data; the velocity map created from interpreting thefringing image can be later graphically overlaid with thenon-fringing image.

The summed signal will also give the system spectrum,that is, the combined spectrum of the illumination, tar-get reflectivity and detector power sensitivity. The band-width of the system spectrum is represented by ∆λsys,and is important for determining the fringing behavior ofthe velocimeter.

It will be implied in the descriptions below, althoughnot explicitly shown for clarity, that what ever optical

1-7

1

2

19

20

Fig. 3B

100

12 16

14

102 104 108 110191

2

3

4

Fig. 3A

Figure 3A shows a simple white light velocimeter configura-tion. Figure 3B shows the use of angle of incidence to dis-tinguish the complementary beams reflecting from a target.

manipulations are done to one detecting output, may alsobe done simultaneously to the other complementary de-tecting output, and that these outputs may subsequentlybe subtracted or summed after conversion to a voltage ordigital value. A velocity can be determined using eithercomplementary outputs singly, or both as a differencesignal.

Furthermore, many techniques that have been em-ployed for conventional VISAR velocimeters can be uti-lized for the present invention. For example, the use ofa λ/4 retarder to produce a 90 phase lag between ver-tical and horizontal polarizations is commonly used todetermine the direction (polarity of change) of the fringeshift in conventional VISARs. This technique can also beused in the present invention, although it is redundant inmany cases since the use of broadband illumination in amulti-channel mode can determine fringe shift polarity.One situation where it wouldn’t be redundant is whenthe velocimeter is used with chirped pulse illumination.

2. Practicalities using complementary outputs

In the simplest white light velocimeter configuration(Figure 3A), one of the complementary outputs 2 of thesource interferometer is discarded. This is fundamen-tally necessary to produce the valleys in the source in-terferometer’s power transmission spectra. Without thevalleys, the changing overlap between source and detect-ing spectra will not produce fluctuations in the detectingoutput, and hence no fringes would be produced. Thusin the simplest configuration shown in Figure 3A, whichuses a Michelson source interferometer with 50% reflec-tive beamsplitter, 50% of the source power must be dis-carded.

Figure 3A is a topological diagram only. The beam-splitters 102 and 104 are shown separately for clarity; inactuality, they may simply be a different portion of thesame optical piece. Similarly for beamsplitters 108 and

Flo

w Left CameraVelocity Sensitivity

Right Camera

174172

161

170

172174

166

162160

168

Directions

Fig. 4

Figure 4 shows an embodiment of the white light velocimeterfor measuring fluid flow.

110. Each interferometer (12 and 16) has two comple-mentary outputs (1, 2 and 3, 4). Each also has two com-plementary inputs (only 19 and 20 are shown). Broad-band light from the source enters the source interferome-ter and is evenly divided between the two source outputs1 and 2. In the simplest configuration (Figure 3A), onlyone of these outputs, 1, is used to illuminate the target.In the complex configuration (Figure 3B), both outputsare used. However, the outputs must be distinguishedfrom each other as they reflect off the target. This canbe accomplished by polarization or angle of incidence. Ifpolarization is used, then polarizing beamsplitters can beused for all the beamsplitters. Figure 3B shows the casewhere angle of incidence is used to distinguish the beamsreflecting from the target. The two reflected beams enterthe two complementary inputs 19 and 20 of the detect-ing interferometer. Therefore, 100% of the illuminatingpower is used in the complex configuration.

E. Example application measuring fluid flow

Figure 4 shows an example application of a white lightvelocimeter measuring fluid flow, such as a gas jet seededwith light scattering particles. The arrangement consist-ing of a light source 160 (e.g., a flashlamp), whose lightpasses through a source interferometer 162 and enters athick illuminating fiber 166. Light leaving the fiber iscollimated by a cylindrical lens 168 into a fan-like shapewhich illuminates a thin slab of the jet volume 170. Thelight scattered from particles in the jet is observed by oneor more camera systems 174, each which observe througha detecting interferometer 172. The detecting interferom-eter delays are matched to the source interferometer inorder to produce fringes. The details of the camera sys-tems are left unspecified. They could be one of a varietyof configurations, measuring a point, a line, or an area ofthe target.

The purpose of Figure 4 is to show that 1) if thesource interferometer optics have sufficiently low aber-

1-8

rations such that cτs is the same for all rays within λ/8,a non-imaging system such as a thick multi-mode fibercan be used to communicate the imprinted light from thesource interferometer to the target. 2) In order to mea-sure different velocity components, there can be morethan one detector and the illumination can be laterallyoriented. The velocity component which is sensed by eachcamera is given by the vector sum of two unit length vec-tors: target-camera and target-source. By having twocameras on opposite sides of the target and the illumina-tion to the top, the vertical and transverse componentscan be resolved. With one or more additional camerasmounted out of the plane of Figure 4, the remaining ve-locity component can be resolved.

1. Range of allowed velocities

Let ∆λsys be the system bandwidth, which is definedthe spectral width of light having passed through all theoptical elements and converted to electrical signal by thephotodetector, including target reflectivity, photodetec-tor responsivity and illumination color. The system co-herence length is Λsys ∼ λ2/∆λsys. Partial fringes resultonly when |cτs − cτd| ≈ Λsys/2. Since target velocitycauses a change in effective τs, the range of velocitieswhich produce visible fringes is controlled by the systembandwidth through∣∣∣∣ cτs

1 + 2v/c

∣∣∣∣ < 12Λsys (11)

Equation 11 is for a single detecting channel. Manyof the attractive velocimeter features are implementedby a multi-channel velocimeter. The multiple detectingchannels can be organized by wavelength or by delay τd.In the former case, the individual channel bandwidth isvery narrow, and τd is the same for every channel. Inthe latter, the channel bandwidth is very wide, but τd isdifferent for each channel. These two modes are describedin more detail later.

F. Theory regarding the superimposing condition

The theoretical reason for requiring the superimposingcondition for the interferometers is as follows. Let thecondition where interference is prevented be called inter-fereless. (This could be achieved in concept by blockingone of the two interferometer arms in the case of a Michel-son, or zeroing the partial reflectivity of the mirrors inthe case of a Fabry-Perot.) Let the optical field in the re-gion of an extended incoherent source be S(t, r) and thefield at every point “downstream” through an interfere-less optical system be H(t, r) where r is a position vector.Now, restoring the interference, if the output rays of theinterferometer beams exactly superimpose, so that theimages of the source viewed through the interferometer

superimpose both transversely and longitudinally, thenwe say the superposition condition is satisfied. In thecase of a multiple echo interferometer such as a Fabry-Perot, this condition must apply to each echo.

If this condition is satisfied for the source interfer-ometer 12, then the velocimeter can be equivalently beexpressed as an interfereless system having an effectivesource S(t, r) + S(t + τ, r), which produces an opticalfield everywhere downstream H(t, r)+H(t+ τ, r). If thedetecting interferometer 16 also superimposes, then thevelocimeter output corresponds to H(t, r)+H(t+τs, r)+H(t+τd, r)+H(t+τs +τd, r). The point is that these ex-pressions are easy to analyze since they all share the samespatial dependence- the same as an interfereless system,which is covered by traditional optics.

Thus in the superimposing case, the spatial dimensionscan be ignored and only the temporal behavior need besolved. The problem collapses to 1-dimension, as if thesignals were propagating along wires. This greatly sim-plifies the analysis.

If the superposition condition is violated, then thesource is described as S(t, r)+S(t+τ, r+δr) where δr isthe displacement vector between the source images seenin the beamsplitter. This produces a complicated opti-cal field downstream since the two terms do not sharethe same spatial dependence. For sufficiently large δr,the fringes at the velocimeter output will wash out andtherefore fail to give a readable velocity signal. Sincesome image displacement is inevitable through imperfectoptics, an estimate is made of the maximum value of δrwhich maintains significantly visible fringes. This will tellus the maximum amount of aberrations we can toleratein the interferometer optics.

1. Distorting elements tolerated

A very practical consequence of the superimposing con-dition is that the path between the target and velocime-ter can contain distorting or dispersive elements, or anon-imaging system such as an optical fiber without ef-fecting the fringe visibility (for a uniform velocity targetand constant distortion). In other words, high qualityoptics are needed only internal to the interferometers.This feature is highly desired for the use of a velocimeteron out of door targets, or communicating light betweenthe velocimeter apparatus and the target through opti-cal fiber, such as in an industrial situation. Once theechoes have been imprinted on the illuminating beam,their phase relationship is maintained, since both echoespropagate over the same path (assuming the distortionchanges slowly compared to τ).

Secondly, the superposition argument above showsthat an imaging velocimeter can be produced if the in-terfereless form of the optics produces an image. Eachpoint of the image has a time averaged intensity depen-dence which is described by the 1-d theory below.

1-9

IV. THREE MODES OF MEASURING FRINGES

There are three modes of detecting the fringes andinterpreting them into velocity: 1) single-channel; 2)multi-channel delay-distributed; and 3) multi-channelwavelength-distributed. The three are mathematicallyrelated. Many of the attractive velocimeter features, suchas simultaneous multiple velocity detection, only comefrom the use of the multi-channel modality. The single-channel modality must be understood to understand themulti-channel modality. All these modalities apply tointerpreting the velocity of the target at a single targetpoint. For a line or areal imaging velocimeter, the modal-ities apply to each pixel of the image.

A. Single detecting channel

In the case of a single detecting channel, broadbandillumination will produce a velocimeter with a narrowvelocity range. From Equation 11, the center of the ve-locity range is given by

vcenter =τs − τd

τs

( c

2

)(12)

and the range width is

vwidth =Λsys

cτs

( c

2

)(13)

Thus, a broad spectrum produces a narrow Λsys, whichin turn yields a narrow range of velocities that give visiblefringes.

This narrowness can be used as a velocity selector ordiscriminator. By numerically subtracting the comple-mentary detecting interferometer outputs after detection,only the fringing targets will produce a nonzero differ-ence signal. This is particularly useful for an imagingvelocimeter, since this means only the objects having aspecific velocity component will appear different from thebackground image. For example in a radar or LIDAR ap-plication, if there are a multitude of missiles approach-ing, and one is only interested in detecting the missileshaving a velocity component indicating that they will im-pact somewhere important, then the operator would setthe interferometer delays according to Eq. 12 so that onlythose velocities produce fringes.

B. Multiple detecting channels

Velocimeters implemented with multiple output chan-nels, organized either by delay or wavelength, offer manyattractive features. The advantages include a muchgreater velocity range for a given overall bandwidth, inde-pendence from target and illumination color, an absolutevelocity determination, and the ability to detect simul-taneous multiple velocities. The latter two abilities are

very important, particularly in the field of shock physicswhere discontinuous accelerations and disintegrating ma-terials create difficult targets. The ability to determineabsolute velocity from a target acquired after it has accel-erated (suddenly appearing at the horizon) is very impor-tant for LIDAR and radar velocimetry, as is being ableto discriminate targets against stationary clutter such aschaff.

1. Wavelength distributed system

A wavelength distributed system is shown in Fig-ure 5A. This consists of the white light velocimeter ofFigure 1A with the added prism 90 to disperse the out-put according to wavelength across a set 92 of multipledetecting channels. In a single target-point mode, thepoint is dispersed into a line which can be shown on theslit of a streak camera to be recorded. In a line-targetmode, the line is dispersed into a rectangle which canbe recorded by a multi-channel areal detector such asa CCD array, or film. Since a rectangle cannot be dis-persed into a third dimension, a two-dimensional imageof the target can be measured if one can uses multiplediscrete two-dimensional detectors, each with bandpassfilters to limit the wavelengths detected. Even a modestnumber of channels, such as 3 or 4, provide advantages,particularly if the channel center wavelengths are widelyseparated.

The greatest advantages of this wavelength-organizedsystem will be when there are many channels individuallyhaving a very narrow bandwidth ∆λsys, but collectivelyhaving a wide bandwidth. When each channel’s ∆λsys

is very small, then individually each channel has a verylarge velocity range according to Eq. 12. That is, for anaccelerating target many fringes can pass before their vis-ibility diminishes. This allows a small velocity per fringecoefficient η, which increases the velocity resolution.

Figure 5B shows the simulated spectrum for a singletarget point versus velocity for a fixed source-detector de-lay difference. The vertical scale is proportional to 1/λor light frequency ω. The dark bands represent fringemaxima or minima, depending on which complementarydetecting output is observed. The plot for negative veloc-ities is a reflection of the positive side. For a given targetvelocity, the number of fringes counting along a verticalslice of the graph will indicate the velocity. This canbe mathematically accomplished by taking the Fouriertransform of the intensity along the vertical slice. A givenspatial frequency component will correspond to a giventarget velocity. The higher the velocity, the higher thespatial frequency.

If more than one target having different velocities is inthe velocimeter field of view, then each target will con-tribute a different spatial frequency component. Thesecan be resolved if the velocities are sufficiently differentcompared to η, given by Eq. 1. The greater the am-plitude of that Fourier component, the greater the net

1-10

1/

Velocity or Delay difference

∆ sys10

12 16

14

9092

Fig. 5A

Blue

Red

Fig. 5B

Figure 5A shows a wavelength distributed system. Figure 5B shows the simulated spectrum for a single target point versusvelocity for a fixed source-detector delay difference.

target area moving with that velocity.

2. Delay distributed system

In a delay-distributed multi-channel system, eachchannel has the same system spectrum, which is op-timally very wide, but at a different delay difference(τs − τd). This can be accomplished by tilting one of theinterferometer mirrors so that delay is a function of theoutput’s transverse position on the camera’s film plane.The reason tilting causes this is shown in Figures 7A-7F.Figure 7A shows a Michelson interferometer with per-fectly aligned mirrors 402, 403. From the point of viewof the detector 405, the reflection of the source 10 ap-pears in the beamsplitter 404 to be at two locations 400,401. These are superimposed for the aligned Michelsonin Figure 7B. In Figure 7C, mirror 402 is tilted out ofalignment. This moves apparent position of the source401 so that it does not superimpose with 400 (Figure 7D).Figure 7E shows a camera lens 406 imaging the two twinsource images on to a film plane 407. Different posi-tions along the film correspond to different angle q of raybundles leaving the source twin, and these have slightlydifferent delay difference ∆delay, shown in closeup Fig-ure 7F.

Position across Figure 6A and 6B perpendicular to thefringe comb corresponds to increasing delay difference(τs − τd). Each resolvable column on the Figure parallelto the fringe comb corresponds to a channel. Just as inthe λ-distributed system, the wide bandwidth providesabsolute velocity determination, multiple simultaneousvelocity detection, and independence from details of thesystem spectrum.

Since the channel bandwidth is very wide, fringe visi-

bility maximizes for a very specific delay difference sat-isfying Eq. 12. Integrating over the intensity changes ofFigure 6A shows the center of fringe pattern is at a welldefined location indicated by the white arrow. For thestationary target (Figure 6A), this corresponds to the lo-cation on the image where (τs − τd) = 0. For the movingtarget (Figure 6B), this corresponds to a different loca-tion on the image where (τs − τd) = 0. Thus the shift incenter position of the fringe pattern yields the velocity.

The location of the center of the fringe pattern canbe determined mathematically by Fourier analysis, fromthe phase shift of each component. This also deter-mines the system spectrum, because each color producesa fringe comb component with periodic spacing propor-tional to λ, and amplitude equal to the intensity of thecolor. Thus, even though this modality does not use aprism or diffraction grating, or a wavelength sensitivedetector, the color components and their relative inten-sities can be resolved. (This is the basis for an existinglaboratory spectroscopy technique called Fourier trans-form spectroscopy). Thus, the wavelength and delay-organized multi-channel systems are mathematically re-lated. Furthermore, this means that the chirped pulsestreaking technique described later can be accomplishedwithout a diffraction grating, if the detecting interfer-ometer is operated in the delay-organized multi-channelmode.

3. Narrow-band illumination

Laser and other narrowband illumination can be ap-plied to the white light velocimeter in the same manneras broadband light. However, the attractive features ofabsolute velocity determination and simultaneous multi-

1-11

0th fringe

a)0 m/s

0th fringe

b)16 m/s

Symmetry

Symmetry

Figure 6A depicts an experimentally measured fringe patternof a stationary object. Figure 6B depicts a an experimentallymeasured fringe pattern of a moving target.

ple target detection are not available for monochromaticlight. Some degree of these features can be retained ifat least 2 or 3 colors are present, such as in a multi-linelaser, i.e., Argon or Krypton ion laser. The user shouldbe aware that if cτs is smaller than the laser coherencelength, (so that the interferometer free spectral range islarger than the laser bandwidth), slight adjustment of τs

may cause the power to switch from one source interfer-ometer output to the other. This is particularly relevantwhen using a multi-line laser, since τs should be selectedsuch that the important spectral lines emerge from thesame output arm.

4. Chirped illumination

An interesting application of the white light velocime-ter is with specially prepared light called chirped light.This embodiment is called a chirped pulse streakingvelocimeter and it can measure velocity or reflectivitychanges versus time for every place on a line imagedacross the target. The device is interesting because itcan potentially reach very short time resolutions, downto 2 ps, depending on the amount of chirp on the light.Let us describe “chirped” light as light which instanta-neously has an approximately pure color which varies

402

403

405

10

404

Fig. 7A

400,401Fig. 7B

402

10

Fig. 7C

400Fig. 7D 401

406 407

400

401

∆delay

Fig. 7E

Fig. 7F

400

401

∆delay

Figures 7A–7F illustrates the effect tilting a mirror has ondelay.

with time. Chirped light is distinguished from incan-descent broadband light in that there is never simulta-neously two different colors present– the colors arrive insequence. Chirped light is usually in pulsed form, but itcould be a continuous beam if the frequency was scannedin a repetitive fashion. This could be produced by a CWor pulsed laser whose wavelength is caused to sweep byan intracavity electro-optic device, or by taking a veryshort laser pulse and sending it through a matched pairof diffraction gratings to stretch it. These techniques arecommon in laser laboratories.

The chirped pulse streaking velocimeter is obtainedsimply by using the white light velocimeter in a multi-channel mode with chirped illumination. Both thewavelength-organized and delay-organized multi-channelmodalities can be used, since they are mathematically re-lated. However, it is easiest to understand the operationof the device when it is in the spectrally organized mode.In that case a line image of the target is imaged to theslit before a prism or grating, which spectrally dispersesthe output. Because the instantaneous color of the lightis changing with time, the output from the grating willsweep across an integrating recording medium such asfilm or a CCD camera.

The chirped pulse streaking velocimeter of Figure 8consists of chirped pulse input 120, source interferometer122, moving target 124, lens 126, detecting interferometer128, slit 130, dispersive element 132 and film or CCD 134.

1-12

124

126

128

122120

130

132

134

Fig. 8

Time and

Position along target

Figure 8 shows the chirped pulse streaking velocimeter.

It is assumed that the instantaneous wavelength versustime is known by an independent measurement. Thus,the position along the streak record in the wavelength di-rection will correspond to time. The complementary de-tecting interferometer output can be similarly recorded,and the two signals subtracted or added. Fringe shifts inthe difference signal are interpreted into a velocity versustime record using Equation 4. The sum signal yields thereflected target light intensity history.

The reader should note that because chirped light is in-stantaneously monochromatic, the chirp pulse velocime-ter does not have the multiple-target, absolute velocityresolving features of the non-chirped broadband config-uration. If these are desired, the user can operate thedevice in a hybrid configuration simply by using morecomplex illumination with the same optical apparatus.The complex illumination could consist of, for example,3 simultaneous colors, which are all being swept across awavelength range 1/3 as wide as the full spectrum. Thatis, the illumination may start as red, yellow, and blue,and end up as orange, green and violet.

If the velocimeter detecting channels are organized bydetecting delay, such as created by tilting an interfer-ometer mirror, then the output beam will “sweep” in adifferent manner. (The grating is not used in this case.)As the different colors sequentially pass out the velocime-ter, they will create fringe combs having different spatialfrequencies. A one-for-one association can be made be-tween a spatial frequency and a moment in time. Thus,in order to analyze the record, a Fourier transform ofthe recorded intensity is made. The phase shift for agiven spatial frequency yields the instantaneous velocity,and the magnitude of the component yields the targetreflected light intensity.

The optical streaking concept can be applied to createa much simpler device that measures only target reflec-tivities and not velocities. This embodiment is called achirped pulse streaking camera. The two interferometersneeded for velocimetry are discarded. The device sim-ply consists of the grating dispersing the chirped outputand the detector. The time resolution capability of this

device is exactly the same as for the chirped pulse ve-locimeter. Alternatively, only the source interferometeris discarded, and the detecting interferometer is used inthe delay-distributed mode without any grating.

The optical streaking of these inventions is analogousto that in an existing electronic streak camera, exceptthat the sweep speed is not controlled by any electricalbehavior inside the camera, but by the degree of chirp ofthe illumination entering the optics. Electronic streakcameras can suffer from jitter in the triggering time.That is, it takes a finite amount of time from the arrivalof the trigger pulse to the beginning of the high voltageramping (which sweeps the electrons), and this lag timecan be unpredictable in single-shot experiments. The ad-vantage of the optical streaking is that there is no jitterin the device’s operation. The chirped diagnostic illumi-nation can be synchronized by optical delays to the otherlaser beams which may drive the experiment being stud-ied (such as studying the expansion of a laser-createdplasma). Optical delays are controlled by physical dis-tances, which are extremely reproducible from shot toshot. Secondly, apart from the expense of the chirped il-lumination, the chirped pulse streaking velocimeter andcamera are relatively inexpensive optical devices to build,and an arbitrarily large number can be used to simultane-ously observe the same experiment from different angles,all illuminated by the same chirped source.

To estimate the time resolution ∆t, the chirped pulsewas modeled as a series of transform limited elementalpulses of width ∆t concatenated over a duration Tp witha linearly ramping average frequency. Using the uncer-tainty relationship for the elemental pulse we find that

∆t =

√Tp

c(1/λ1 − 1/λ2)= 1.8ps

√Tp(ns)

(1/λ1 − 1/λ2)(µm−1)(14)

where Tp is the pulse duration in ns and λ1 and λ2 are thebracketing wavelengths in µm describing the chirp. Forexample, with Tp=100 ns and a 10% chirp from λ1=500nm to λ2 = 550 nm, Eq. 14 yields ∆t=43 ps. The ratio ofrecord length to time resolution is 100000/43=2300. Thisis an order of magnitude greater than the ratios (∼200)estimated for the phosphor screens of an electronic streakcamera having a ∼30 mm diameter screen. For Tp=10 µsand the same 10% chirp as above, Eq. 14 yields ∆t=0.43ns and a record length ratio of 23000. This vastly exceedsthe capability of today’s electronic streak cameras.

V. INTERFEROMETER DESIGN CLASSES

A. Michelson vs. Fabry-Perot

Interferometer designs can be organized into twoclasses according to the number of output pulses pro-duced from a single applied pulse. A Fabry-Perot-class

1-13

/2

Time

Freq0

1Fig. 9F

T( )

Fig. 9DFig. 9E

Output3436

35

Fig. 9B

Time

Output

Fig. 9A

Freq

T( )0

1

Fig. 9C

30 33

31

32

Figure 9A shows a Michelson class interferometer using anetalon delay. Figure 9B shows the impulse response for theMichelson class interferometer. Figure 9C shows the sinu-soidal power transmission spectrum for the Michelson classinterferometer. Figure 9D shows the Fabry-Perot class inter-ferometer in a superimposing configuration. Figure 9E showsthe impulse response for the Fabry-Perot class interferometer.Figure 9F shows the thinner peaks of the Fabry-Perot classpower transmission spectrum.

(F-P) interferometer generates an infinite series of out-put pulses, whereas the Michelson-class generates a finitenumber (two). It will be shown that the number of out-put pulses determines the number of fringing sidebands inthe intensity versus velocity plot discussed below. Thereis also a fundamental difference in the fringe visibilityand power throughput of the two classes. By increas-ing the partial reflectivity R of the F-P mirrors, the ve-locimeter fringe visibility can be increased arbitrarily atthe expense of smaller throughput. This is not true ofthe Michelson-class, where the fringe visibility cannot ex-ceed 50% for a single output. (Push-pull subtraction willincrease fringe visibility by eliminating constant compo-nents.)

Figures 7A and 7D show these two interferometerclasses, distinguished by their impulse responses. TheMichelson class shown in Figure 9A consists of a beam-splitter 30, mirror 31, and a superimposing delay, whichin this case is the combination of an etalon 33 and mirror32. Other superimposing delay designs described belowcan be substituted. Figures 9B and 9C shows that theimpulse response for the Michelson class is pair of echosseparated by τ .

The Fabry-Perot (F-P) class shown in Figure 9D con-sists of two partially reflective mirrors 34, 35 separatedby a distance τ/2 and having a lens 36 placed in betweenmirrors 34, 35. The lens must image the surface of onemirror on to the other. The lens is crucial for the su-

perimposing behavior. An ordinary F-P does not havethis lens and is not superimposing. For the Fabry-Perot(F-P) class, the impulse response is an infinite number ofechos of geometrically decreasing amplitude, as shown inFigure 9E.

Figure 9C shows the power transmission spectrum forthe Michelson class interferometer is sinusoidal. The F-Pclass power spectrum can have thinner peaks, as shownin Figure 9F, depending on mirror partial reflectivity R.

B. Superimposition

Associated with each output pulse of the impulse re-sponse is an image of the source. The purpose of the su-perimposing interferometer is to superimpose these im-ages as closely as possible, for all wavelengths and in-cident ray angles, while one image is delayed in termsof time-of-flight of the light. This is achieved in prac-tice only for a limited range of wavelengths and incidentangles due to ordinary and chromatic aberrations. Thegreater these ranges, the greater the power available forvelocimetry from non-directional broadband sources suchas incandescent lamps. Thus the chief design goal for asuperimposing delay is to minimize aberrations for therange of wavelengths and ray angles desired.

The dispersion of glass optics affects the interferom-eter two ways: it chromatically smears the image posi-tion which detriments the superposition, and secondly, itcauses a wavelength dependence to the time delay. Thetime delay smear is not a problem. However, the imageposition smear is a problem since it can seriously reducethe fringe visibility. Any other aberrations, chromaticor ordinary, which smear the images will detriment thesuperposition.

1. Tolerance for smear

The tolerance for smear will be estimated. Let thesmear be modeled by a simple displacement vector δrof twin images seen in the beamsplitter in a Michelsondesign. The tolerance between the multiply displacedimages of a superimposing F-P should be similar. It willbe shown that the narrower the numerical aperture of theoptics, the greater δr which can be tolerated.

Let the displacement vector δr be

δr = xδrx + zδrz (15)

with components δrx and δrz in the transverse x, andlongitudinal z (on-axis) directions. By comparing thedistances to two point-sources on an entrance pupil froma single third point on an exit pupil, and requiring thedifference in the two distances to be constant within atolerance λ/4, it can be shown that the maximum dis-placements in the longitudinal and transverse directions

1-14

must be less than

δrz <λ

21θ2

(16)

and

δrx <λ

81θ

(17)

where θ is an angle which characterizes the numericalaperture subtended between the entrance and exit pupils,whichever numerical aperture is larger.

Equations 16 and 17 can be used in reverse, to solvefor the range of angles θ from an illumination sourcewhich produce visible fringes (and therefore suitable forvelocimetry) for a known displacement between twin im-ages in the interferometer. The displacement vector δrwould be caused either by aberrations, or by use of anon-superposing interferometer.

For example, this can be used to show the disadvan-tages of a non-superimposing interferometer. A non-superimposing interferometer will not be able to utilizethe full power available from a non-directional broadbandsource such as an incandescent lamp. Either the band-width will be limited for all ray angles, or the ray angleswill be limited for all bandwidth. An example of this isthe non-superimposing double Fabry-Perot of US Patent4,915,499. In the former case, the patent authors cal-culated in their aforementioned Applied Optics articlethat their bandwidth was limited to <4 nm when theinterferometer is used to measure off-axis fringes, (andtherefore involving a large range of ray angles.) In thelatter case, which is when on-axis fringes are measured,by using Equation 16 where δrz is the mirror spacingof several cm, it can be seen that the range of accept-able angles is very small, of the order θ ∼0.002 radian.Thus the only a superimposing interferometer (and thusδrz ≈ 0) will simultaneously allow a large range of anglesand large bandwidth, and therefore utilize the full poweravailable from an extended area broad bandwidth source.

C. Fabry-Perot Interferometers

Two kinds of interferometers will be discussed: super-imposing Fabry-Perots, and Michelsons incorporating su-perimposing delays.

The reason the classical Fabry-Perot (F-P) interferom-eter is not superimposing is illustrated in Figure 10A. Anoff-axis entrance ray 39 zig-zags between the mirrors 41and 42, producing output rays 40 that are successivelytranslated transversely, for each round trip. To be super-imposing, the output rays must share the same path, forall round trips. For the classical F-P, this only occurs forthe on-axis ray.

In a superimposing Fabry-Perot, shown in Figure 10B,echoes for all the round trips exit along the same out-put ray 44, for each incident ray 43. This condition issatisfied for any incident ray angle, not limited to the

Fig. 10B

43 44

48 46 45 46 49

Fig. 10A 41 42

3940

Fig. 10C 46' 45 47

Fig. 10D50 51

Figure 10A illustrates the nonsuperimposing Fabry-Perot in-terferometer. Figure 10B illustrates a superimposing Fabry-Perot. Figure 10C shows a superimposing relay delay usingreal imaging. Figure 10D illustrates the path of a ray throughan etalon delay.

on-axis rays as in the classical F-P. Thus, the temporalcharacteristics of the superimposing F-P is the same asthe on-axis classical F-P.

The superimposing F-P is achieved by a lens 45 be-tween two partially reflecting spherical mirrors 46 and47, such that exactly +1 magnification is achieved perround trip. This is achieved when the surface of mirror46 is imaged to the surface mirror 47 by the lens 45. Theradii of curvature of the spherical mirrors are centered atlens 45. Lenses 48 and 49 allow a collimated beam to passas collimated. The interferometer will be achromatic ifan achromatic lens 45 is used.

D. Superimposing delays

Superimposing delays which can be incorporated intoone arm of a Michelson interferometers will now be de-scribed. Generally, a superimposing delay should act likea plane mirror in terms of how the output ray directionsbehave, but yet delay the light in time. The createdmirror will be called the “effective” mirror. (The term“virtual” is avoided here to prevent confusion with theterm virtual imaging, used below to classify the lens ar-rangement.) The mirror of the other interferometer armis positioned to superimpose with this effective mirrorposition, thus creating a delay while satisfying the su-perposition condition.

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63

4861

46' 45 47

62

Fig. 11

Figure 11 shows how a superimposing delay is used in onearm of a Michelson interferometer.

1. Etalon delay

The simplest superimposing delay is the etalon delay.Figure 10D shows how due to refraction the mirror 51behind the glass slab 50 appears closer to the slab, yeta pulse is delayed by the slower speed of light throughthe glass. The etalon delay is used in conventional laser-illuminated VISAR velocimeters, as shown in Figure 9A.The advantage of the etalon is its simplicity of alignment.However, the etalon delay suffers chromatic aberrationbecause of dispersion in the glass. Thus it can only beused in the white light velocimeter for high collimatedlight, or for sufficiently small delays such as for high ve-locity measurements.

The other superimposing delays described below areachromatic. The achromatism comes from the use of indi-vidually achromatic lenses and mirrors, or the matchingof elements having opposite chromatisms. Except wherenoted, the mirrors are assumed to be first surface reflec-tors and therefore achromatic. To avoid duplication, thedrawings will chiefly show the use of transmissive optics.However, it is implied that reflective optics could option-ally be substituted with minor rearrangement of positionsto enhance the achromatic quality or reduce cost.

2. Delays using real imaging

The delay designs can be classified according towhether real or virtual imaging is used between a lensand a mirror. This strictly depends on the lens-mirrordistance compared to the focal length.

A relay lens delay is a superimposing delay using realimaging related to the superimposing F-P. Referring toFigure 10C, the relay lens delay is obtained from thesuperimposing F-P optics, by making the exit mirror 47totally reflective and removing the entrance mirror. Lens45 images the mirror 47 surface to the plane at lens 46′.The focal point of lens 46′ is at lens 45, and the center ofcurvature of mirror 47 is at lens 45. These two arrange-ments allow a collimated beam to pass through the delayas collimated. The optics as a unit act as a plane mirrorsuperimposed at the position of lens 46′, and yet it takeslight a finite time to traverse the unit. Lenses 45 and 46′should be achromatic to produce an achromatic delay.

Figure 11 shows how this delay is used in one arm of

10

65

48

46'66

4768

61

50

69

70

62

63

Fig. 12

Figure 12 shows an embodiment of a complete velocimeterusing retro-reflected light.

a Michelson interferometer, as the group of elements en-closed in the dashed box 62 form an effective plane mirrorat 48. The mirror of the other arm 61 is superimposed bybeamsplitter 63 at this plane 48. To implement a meterscale delay having f/10 optics, the internal lens 45 canbe implemented by a spherical mirror 65, as shown inFigure 12. This shows a complete single interferometervelocimeter including source 10, target 69 and detectingtelescope or camera 70. Lens 50 images the target 69to the interferometer mirror planes 48 and 61. The pur-pose of this is discussed later. The optics enclosed by thedashed box 62 form an effective plane mirror at 48. Thepick-off mirror 66 is needed so that lens 46′ and mirror47 are the same distance from mirror 65. To minimizeastigmatism due to off-axis reflection from the sphericalmirror 65, the rays 68 should pass closely to the pick-off mirror 66. On the other hand, the size of the imageof the target at plane 48, which is also the size of theimage at mirror 47, limits the rays 68 from coming veryclose to mirror 66. Thus there is a compromise betweenastigmatism and image size.

3. Lens delays using virtual imaging

Figure 13A shows a virtual imaging delay using a com-bination of an achromatic transmissive lens 210 and anfront surface spherical mirror 212 (which is achromatic).In this design, lens 210 and mirror 212 are separated by adistance less than the lens’s focal length F (213), hence,virtual imaging is involved. If the lens is positive, themirror is convex as in Figure 13A; if the lens is negative,the mirror is concave as in Figure 13C.

Figure 13A and 13C, the focal point of lens 210 is ar-ranged to be at the mirror center of curvature 214 so thatthe lens 210 images point 214 infinitely far way. For thecase of a positive lens 210 in Figure 13B, the surface ofthe mirror at point 216 is imaged by lens 210 to a po-sition 217. Thus, the effective image of mirror 212 is aplane mirror 213 at a position 217. Similarly for the case

1-16

Fig. 13A

Fig. 13B

Fig. 13C

213

210 212

210

212

213

216 217

210 212

214

214

Figure 13A shows a virtual imaging delay using a combina-tion of an achromatic transmissive lens and a front surfacespherical mirror. Figure 13B shows the surface of a mirrorimaged by a lens to a fixed position. Figure 13C shows theuse of a negative lens and a concave mirror.

of a negative lens 210.

4. Minimizing chromatic aberrations

In Figure 14, if an etalon 240 is placed in between thelens 241 and the reflector 242, the chromatic aberrationintroduced by the etalon is of opposite polarity as thatof a simple lens and therefore can reduce or zero thenet aberration of the combination lens-etalon. The raysdrawn in Figure 14A are drawn for blue and red and thedispersive quality of the etalon is greatly exaggerated.The rays are drawn as if the refractive index for red is1 (i.e., no refraction) and for blue much larger than 1(much refraction). An object behind a slab of glass al-ways appears closer to the observer. Since the refractiveindex is greater for blue than red, the mirror center ofcurvature 244 will appear closer to the lens for blue raysthan red rays. This may help compensate the chromaticaberration of a simple uncorrected lens 241, since thatlens will focus the blue closer to lens 241 than for red.Thus, the apparent mirror center of curvature will remainat the focal point of the lens for both blue and red.

Fig. 14A

241240 242

Fig. 14B

241 245 246 242

Figure 14A shows the use of an etalon to reduce or zero the netaberration of the combination lens-etalon. Figure 14B showsthat tilting glass etalons can adjust the amount of chromaticaberration contributed by the etalons.

Figure 14B shows that tilting glass etalons 245 and246 can adjust the amount of chromatic aberration con-tributed by the etalons, so that the net amount for thecombination lens-etalons can be zeroed. A pair, insteadof a single etalon, may be used to prevent introductionof other aberrations. One may need an additional pair oftilted etalons (not shown) tilted in an orthogonal direc-tion (out of the plane of the Figure) to avoid astigmatism.

5. Creating negative and positive delays

The introduced delay can be positive or negative de-pending on the total glass thickness in the delay, and theposition of the effective mirror 242′. In practice, onlythe magnitude of the delay is relevant, since moving thedelay to the other arm will flip the polarity, or swappinginterferometer outputs will have the same effect.

In Figure 15A, a thin achromat 250 and small radius ofcurvature mirror 252 create a large negative delay, sincea photon traveling through the combination will appearsooner than a photon traveling to the apparent mirrorand back. This is because the apparent mirror 252′ isfar to the right, and the amount of glass small. In Fig-ure 15B, a thick achromat 254 and large radius of curva-ture mirror 256 create a positive delay. A photon travel-ing through the combination will emerge later than onetraveling to the apparent mirror 256′, because the lighttravels slower through glass and the apparent mirror isclose. Figure 15C shows a design with a negative lens260 that achieves a wider angle of acceptance then 13Afor the same large delay.

1-17

Fig. 15A252250 252'

Fig. 15B

254256

256'

Fig. 15C

260

Figure 15A illustrates the use of a thin achromat and a smallradius of curvature mirror to create a large negative delay.Figure 15B illustrates the use of a thick achromat and a largeradius of curvature mirror to create a positive delay. Figure15C shows a design with a negative lens that achieves a widerangle of acceptance then 13A for the same large delay.

D1

SA

SA'

R

F

270270'

D3D2Fig. 16

Figure 16 illustrates the variables used to compute net delay.

6. Computing the net delay

Figure 16 illustrates the variables used to compute thenet delay. We must have F = R + Sa in order for theapparent mirror to be plane (have infinite radii of cur-vature.) The delay is the sum of two components: delaydue to the slower speed through the glass, and delay dueto the change in distance between actual 270 and appar-ent 270′ mirror positions. Let the former delay be calledτglass and the latter τmirror.

274

271 272 273

276

277 278

Fig. 17

Figure 17 shows the use of an imaging optical waveguide.

The insertion delay of a glass element of thickness D is(n − 1)D, where n is the refractive index. Each piece ofglass contributes a delay, so to sum over all the elements

τglass =∑

Di(ni − 1) (18)

where ni and Di are the refractive index and thicknessat the axis of each element of a compound lens.

The apparent mirror distance is computed from thelens law. Let the distance from the lens’ principal planeto the mirror surface at the axis be Sa, and to the imagedmirror surface S′

a. Thus

1S′

a

=1Sa

− 1F

(19)

Equation 19 was written so that S′a is a positive quan-

tity. Then

τmirror = S′a − Sa =

11/Sa + 1/F

− Sa (20)

The net delay is

τ = τglass − τmirror (21)

By using telescope mirrors, achromatic superimpos-ing delays of the order several meters long can be con-structed, having reasonable large numerical apertures.However, larger delays would require prohibitively largemirrors to maintain the same numerical aperture. Onesolution is to use single mode fiber optics to create a de-lay. The delay is superimposing since it only supportsone ray direction. Unfortunately, this precludes imagingvelocimeters, which are desirable.

7. Use with optical waveguide

Another solution is to use an imaging optical waveguide, shown in Figure 17. This consists of a thick (sev-eral mm) “fiber” 270 with a refractive index that de-creases approximately as the radius squared. In such awave guide, real images are formed periodically (271, 272,and 273) at specific distances along the fiber, as shownin Figure 17, similar to a relay chain of positive lenses.These wave guides are commercially available. If a single

1-18

72 70

78

77

76 81,82 80

72 70Fig. 18

Figures 18A and 18B illustrate why imaging the target to theinterferometer mirror plane improves image superposition foraberrated optics.

mirror is placed at the end of the fiber, the mirror is im-aged to the front neighborhood 274 of the fiber and thusforms a superimposing delay for monochromatic light.The glass dispersion prevents its use with broadbandedlight with a single mirror.

To use the wave guide fiber for broadband light a dis-tributed mirror 276 is needed, where the mirror positionfor each color (277 for blue, 278 for red) is different sothat the image of the mirror at the front of the fiber 274is superimposed for all colors. The distributed mirrorwould have a large thickness, of the order (nblue−nred)D,where nblue,red are the refractive indices at the center ofthe fiber for blue and red light, and D is the fiber length.For typical glass, (nblue − nred) is of the order one per-cent. Thus for example, if the fiber is 10 meters long, thedistributed mirror will be approximately 10 cm.

A distributed mirror could be constructed of a vol-ume hologram. This operates on the principle that atransparent material which has a refractive index whichvaries sinusoidally will strongly reflect light of the wave-length matching the periodicity of the sinusoidal varia-tion. Standard holographic techniques can be used toconstruct the hologram such that the effective positionof the reflector varies in depth with color.

8. Why put image at mirror plane

In all realistic superimposing delay designs, some aber-rations are inevitable and these will cause the effectivemirror of the delay to have a slight curvature, instead ofbeing ideally plane. When the plane mirror of the non-delayed interferometer arm and effective mirror are su-perimposed by the interferometer beamsplitter, any cur-vature of the effective mirror can detriment fringe visi-bility. Such a detriment can be significantly reduced ifthe target is imaged close to the mirror planes, and notin front or behind it. In Figure 12, this is accomplishedby the lens 50. The improvement in fringe visibility bythe use of this lens can be dramatic.

Figure 18A illustrates why imaging the target to theinterferometer mirror plane improves image superposi-tion for aberrated optics. The effective mirror positionsof the two arms of the detecting interferometer, delayed

and non-delayed, are located at 70 and 72 after beingsuperimposed by the beamsplitter. Suppose the effectivemirror at position 72 has curvature due to distortions.Suppose also that a lens images the target to position76 far from mirror positions 70 and 72. Then any mir-ror curvature greatly changes the magnification of onereflection 77 of the target image compared to the other78. This increases the displacement vector δr betweenthe twin images, which needs to be minimized to achievevisible fringes. However, if the target is imaged to posi-tion 80 (Figure 18B) near the mirror surfaces, no changein magnification is produced between its twin reflections81 and 82.

9. Use with sound or microwaves

The delays and interferometers discussed have manip-ulated waves such as light traveling as rays in more thanone spatial dimension. The same 3-dimensional configu-rations of lenses and reflectors can apply directly to otherdirectional waves, such as sound and microwaves, pro-vided the size of the elements is appropriately large sothat diffraction of the waves is not serious. If the elementsize is small compared to the wavelength so the diffrac-tion is relevant, then the shape of the elements may beslightly different.

10. One-dimensional configurations

If the wave travels along only one path (single-modepropagation), instead of a manifold of paths, then thesuperimposing requirement is moot. For example for mi-crowaves and sound, an antenna or transducer can sam-ple the 2 or 3-dimensional wavefield and produce a time-varying voltage. The voltage wave then propagates alongan electrical cable in a single mode. A lens focusing lightonto a single-mode optical fiber is another example.

For a single-mode wave, the superimposing interfer-ometer simply consists of delaying a portion of the wave,and then recombining it. For an electrical wave, this canbe implemented by analog delay lines, or done digitallyafter the voltage is converted to a number by an ana-log/digital converter. For a single-mode optical fiber, theinterferometer can be implemented by fiber loops usingappropriate single mode couplers acting as beamsplitters.These 1-D interferometers cannot form prompt imagingvelocimeters. However, by scanning the beam from theantenna or transducer, an image can be constructed overtime.

A single-mode fiber-optic interferometer can have enor-mous delay lengths not practical by use of open-air op-tics. This creates cm/s velocity sensitivity. In order toautomatically compensate for fluctuations in fiber lengthdue to temperature and physical vibrations, the reflectedlight from the target could be retro-reflected so that the

1-19

source and detecting interferometers are implemented bythe same fiber.

11. For matter waves

Crystals and other structures having periodicity ona small length scale are known to interact with matter(such as electrons, neutrons, other sub-atomic particles,atoms and small molecules) as though they were waves.Thus, in principle it is possible to construct an inter-ferometer for matter waves. Two such interferometersin series topologically and having nearly matched delayscould measure the change in energy spectrum of wavesreflecting off or transmitting through a target region be-ing probed. Such a device could form a sensitive particleenergy meter able to detect very slight energy changes.These could be caused, for example, by gravitational orelectric fields, or absorption of energy when the particlebeam interacts with the target, or emits other particles.The change in energy spectrum of the matter waves istreated by the invention analogous to the Doppler changein energy spectrum of light previously discussed.

VI. EXAMPLE APPARATUS USINGRETRO-REFLECTED LIGHT

Figure 12 shows the apparatus of a demonstrationwhite light velocimeter measuring a 16 m/s target usingordinary incandescent light. The color fringe output of itis shown in Figure 6A and 6B for stationary and movingtarget, respectively. A single Michelson interferometerwith a superimposing relay lens delay (the optics in thedashed box 62) is used. By using the retro-reflected lightfrom the target, the single interferometer accomplishesboth source and detecting interferometer functions. Thissimplifies alignment and automatically insures the sourceand detecting delays match closely.

In order to measure the 16 m/s speed, the velocityper fringe coefficient was arranged to be η ≈ 19 m/s perfringe, by a delay of cτ=4 m. This delay is notably longerthan that of a typical VISAR, which commonly has de-lays of only a few centimeters. Lens 50 images the target64 to the interferometer mirror planes. This is importantfor good fringe visibility, since the effective mirror sufferssome astigmatic distortion due to the off-axis reflectionfrom the spherical mirror 52. The illumination source10 was a 10× microscope objective roughly collimatingthe output of a 600 µm diameter fiber illuminated by aincandescent lamp. The target 69 is a fan with reflectivetape on its blades, and a telescope or camera 70 is placedto receive the exit light traveling from beamsplitter 63.Mirrors 61 and 66 are plane mirrors, and mirrors 65 and47 have a 1 m radius of curvature and are separated 1 m.Lens 46′ is a 1 m focal length lens. The other interferom-eter output (not used here) travels toward the source and

is accessible by inserting a beamsplitter between beam-splitter 63 and source 10.

Figures 6A and 6B show the fringe output of the in-terferometer. The beamsplitter or interferometer mirroris slightly tilted to create a comb of fringes, so that thedelay difference τs-τd varies transversely across the filmrecord. The figures compares the fringes obtained for thestationary target to the fringes of the target moving at16 m/s. The white arrows indicate the central fringe ofthe pattern. A shift in the fringe pattern is observed thatis proportional to the target velocity.

The velocity is determined from the shift measured interms of fringe spacing, which in turn depends on color.All of the colors are present and each create a fringecomb. Any of them can be used for the velocity determi-nation. Observing through a 500 nm bandpass filter, ashift of ∆φ = 0.85±0.07 fringe was observed. For λ=500nm, η=18.7 m/s from Eq. 4, a interferometrically mea-sured velocity of 15.9± 1 m/s was measured. A strobo-scope measuring rotational frequency, and measurementof the range of radii which the beam intercepted the bladeyielded a velocity component toward the velocimeter of15.7± 0.3 m/s.

This demonstration measured the target velocity aver-aged over an area defined by the illuminating beam cross-section on the target, because the camera was greatly de-focused. To create a velocity image of the target, ratherthan measuring a single value velocity, the camera wouldbe nearly focused, forming an image of the target onthe film plane. The camera should not be perfectly fo-cused unless the interferometer is perfectly aberration-less. Given that there is inevitably some displacementδr2 between the twin images of the target on the filmplane, the camera must be defocused enough that the fo-cal blur exceeds δr2, so that the two separate focal spotsassociated with pixels from each target twin image over-lap.

Changes and modifications in the specifically describedembodiments can be carried out without departing fromthe scope of the invention, which is intended to be limitedby the scope of the appended claims.

Acknowledgments

This work was performed under the auspices of theU.S. Department of Energy by the University of Cali-fornia, Lawrence Livermore National Laboratory undercontract No. W-7405-Eng-48.

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VII. CLAIMS

(Submitted, not official version)1. A method for generating partial fringes, comprising:

generating broad bandwidth illumination comprising a plural-ity of originating rays denoted white waves irrespective of actualwavelength and medium;

imprinting at least one coherent echo on said white waves toproduce an imprinted beam, wherein each echo of said at least onecoherent echo is superimposed in ray direction and position withthe ray from which each said echo originated, wherein each echo ofsaid at least one coherent echo is separated in time by a delay τs

with respect to each other and to said white waves;

reflecting said imprinted beam from an object to produce a re-flected beam comprising a plurality of reflected rays;

imprinting at least one second coherent echo on said reflectedbeam to produce a second imprinted beam, wherein for each re-flected ray of said reflected beam, each echo of said at least onesecond coherent echo is superimposed in ray direction and positionwith the reflected ray from which said echo originated, whereineach echo of said at least one second coherent echo is separated intime by a delay τd with respect to each other and to said reflectedbeam; and

detecting the time-averaged intensity of said second imprintedbeam; wherein the average is determined over a time at least aslong as the coherence time of said white waves; wherein partialfringes occur in said intensity of said second imprinted beam whenτsτd, within a coherence length.

2. The method of claim 1, further comprising:

repeating the above steps to produce a second set of partialfringes;

taking the phase difference between said partial fringes and saidsecond set of partial fringes to produce a fringe shift; and

multiplying said fringe shift by a proportionality constant toproduce a quantity selected from a group consisting of velocity anddisplacement.

3. A method for measuring the velocity or displacement of anobject, comprising:



with respect to each other and to said white waves;

reflecting said imprinted beam from an object to produce a re-flected beam comprising a plurality of reflected rays;

imprinting at least one second coherent echo on said reflectedbeam to produce a second imprinted beam, wherein for each re-flected ray of said reflected beam, each echo of said at least onesecond coherent echo is superimposed in ray direction and positionwith the reflected ray from which said echo originated, whereineach echo of said at least one second coherent echo is separated intime by a delay τd with respect to each other and to said reflectedbeam;

detecting the time-averaged intensity of said second imprintedbeam; wherein the average is determined over a time at least aslong as the coherence time of said white waves, wherein partialfringes occur in said intensity of said second imprinted beam whenτsτd, within a coherence length;


taking the phase difference between said partial fringes and saidsecond set of partial fringes to produce a fringe shift; and

multiplying said fringe shift by a proportionality constant toproduce a quantity selected from a group consisting of velocity anddisplacement.

4. The method of claim 1, wherein the step of detecting thetime-averaged intensity further comprises distributing detection ofsaid time-average intensity into multiple detecting channels.

5. The method of claim 4, wherein said multiple detecting chan-nels are organized by wavelength.

6. The method of claim 4, wherein said multiple detecting chan-nels are organized by imprinted delay τd.

7. The method of claim 4, wherein said white waves comprisechirped waves having an instantaneous wavelength that changesas a function of time, in conjunction with said multiple detectingchannels.

8. The method of claim 4, further comprising measuring thetime dependence of target reflectivity with chirped waves compris-ing illumination having an instantaneous wavelength that changesas a function of time, in conjunction with said multiple detectingchannels.

9. The method of claim 1, wherein said broad bandwidth illu-mination comprises periodic waves.

10. The method of claim 9, wherein said periodic waves areselected from a group consisting of electromagnetic radiation waves,pressure waves, sound waves and matter waves.

11. The method of claim 1, wherein said second imprinted beamcomprises complementary output beams, said method further com-prising subtracting intensities of said complementary output beamsfrom each other.

12. The method of claim 1, wherein said second imprinted beamcomprises complementary output beams, said method further com-prising adding intensities of said complementary output beams toeach other.

13. The method of claim 4, further comprising forming an imageof said object on a detector of said multiple detecting channels.

14. The method of claim 13, wherein said image is selected froma group consisting of a line-image and a real image.

15. An apparatus for generating partial fringes, comprising:means for generating broad bandwidth illumination comprising

a plurality of originating rays denoted white waves irrespective ofactual wavelength and medium;

means for imprinting at least one coherent echo on said whitewaves to produce an imprinted beam, wherein each echo of saidat least one coherent echo is superimposed in ray direction andposition with the ray from which each said echo originated, whereineach echo of said at least one coherent echo is separated in time bya delay τs with respect to each other and to said white waves;

means for reflecting said imprinted beam from an object to pro-duce a reflected beam comprising a plurality of reflected rays;

means for imprinting at least one second coherent echo on saidreflected beam to produce a second imprinted beam, wherein foreach reflected ray of said reflected beam, each echo of said at leastone second coherent echo is superimposed in ray direction and posi-tion with the reflected ray from which said echo originated, whereineach echo of said at least one second coherent echo is separated intime by a delay τd with respect to each other and to said reflectedbeam; and

means for detecting the time-averaged intensity of said secondimprinted beam; wherein the average is determined over a timeat least as long as the coherence time of said white waves; whereinpartial fringes occur in said intensity of said second imprinted beamwhen τsτd, within a coherence length.

16. The apparatus of claim 15, wherein said means for gener-ating broadband illumination comprises a source of waves whosebandwidth is greater than 1/τs, wherein said source comprises aset of individually narrow bands if the overall bandwidth of the setis greater than 1/τs.

17. The apparatus of claim 16, wherein said source is selectedfrom a group consisting of incandescent lamps, the sun, explosivecompression of gases, electric discharges, gas discharge lamps, fluo-rescence, flames, light emitting diodes, diode lasers, lasers operatedsimultaneously in more than one wavelength, combinations of laserswhich individually are monochromatic, short pulse lasers, chirpedlasers.

1-21

18. The apparatus of claim 16, wherein said source is selectedfrom a group consisting of vibration and any broad bandwidthsound.

19. The apparatus of claim 16, wherein said source is selectedfrom a group consisting of noise, explosions, impulsive pressurescreated from laser irradiation and sound created from a transducer.

20. The apparatus of claim 16, wherein said source comprisesbroad bandwidth electromagnetic waves selected from a group con-sisting of radio waves, microwaves, ultraviolet lamps, x-rays sourcesand broad bandwidth matter waves selected from a group consist-ing of beams of electron, neutrons, alpha particles and atoms.

21. The apparatus of claim 15, wherein each echo of said at leastone coherent echo is superimposed in ray direction and positionwith the ray from which each said echo originated, wherein eachecho of said at least one coherent echo is separated in time by adelay τs with respect to each other and to said white waves.

22. The apparatus of claim 15, wherein said means for imprint-ing at least one coherent echo comprise a superimposing interfer-ometer selected from a group consisting of a superimposing Fabry-Perot and a superimposing Michelson, wherein said Fabry-Perotcomprises interferometers creating an infinite number of echos ofdecreasing intensity for a given applied pulse, wherein said Michel-son comprises interferometers creating a finite number of echos for agiven applied pulse, including variants such as the Twyman-Green,Mach-Zehnder and Sagnac arrangements, wherein said Fabry-Perotfurther comprises means for imaging the surface of one partiallyreflective mirror to another imaging lens or spherical mirror inter-nal to the partially reflective mirror cavity, wherein said Michelsonfurther comprises a superimposing delay in one interferometer arm,said superimposing delay comprising optical elements selected froma group consisting of combinations of lenses and curved and planemirrors, optical fibers with graded refractive index and distributedmirrors.

23. A method for measuring the change in energy spectrum ofwhite waves interacting with a potential-field or matter in a regionof the target volume, which could be free-space, comprising:



with respect to each other and to said white waves;transmitting said imprinted beam through a volume to produce

a transmitted beam comprising a plurality of transmitted rays;imprinting at least one second coherent echo on said transmit-

ted beam to produce a second imprinted beam, wherein for eachtransmitted ray of said transmitted beam, each echo of said at leastone second coherent echo is superimposed in ray direction and po-sition with the transmitted ray from which said echo originated,wherein each echo of said at least one second coherent echo is sep-arated in time by a delay τd with respect to each other and to saidtransmitted beam;

detecting the time-averaged intensity of said second imprintedbeam; wherein the average is determined over a time at least aslong as the coherence time of said white waves; wherein partialfringes occur in said intensity of said second imprinted beam whenτsτd, within a coherence length,


taking the difference between said partial fringes and said secondset of partial fringes to produce a fringe shift; and

converting said fringe shift into a change in energy spectrum ofsaid white waves.

24. An apparatus for measuring the change in energy spectrumof white waves interacting with a potential-field or matter in aregion of the target volume, which could be free-space, comprising:

means for generating broad bandwidth illumination comprisinga plurality of originating rays denoted white waves irrespective ofactual wavelength and medium;

means for imprinting at least one coherent echo on said whitewaves to produce an imprinted beam, wherein each echo of saidat least one coherent echo is superimposed in ray direction andposition with the ray from which each said echo originated, whereineach echo of said at least one coherent echo is separated in time bya delay τs with respect to each other and to said white waves;

means for transmitting said imprinted beam through a volume toproduce a transmitted beam comprising a plurality of transmittedrays;

means for imprinting at least one second coherent echo on saidtransmitted beam to produce a second imprinted beam, whereinfor each transmitted ray of said transmitted beam, each echo ofsaid at least one second coherent echo is superimposed in ray di-rection and position with the transmitted ray from which said echooriginated, wherein each echo of said at least one second coherentecho is separated in time by a delay τd with respect to each otherand to said transmitted beam; and

means for detecting the time-averaged intensity of said secondimprinted beam; wherein the average is determined over a timeat least as long as the coherence time of said white waves; whereinpartial fringes occur in said intensity of said second imprinted beamwhen τsτd, within a coherence length.

IL-9745BWLVcont1.tex 12/03

Part 2

White Light VelocityInterferometer,continuedThis is a continuation of US Patent 5,642,194. Only theclaims section is different.

US Patent 5,910,839Issued June 8, 1999


The invention is a technique that allows the use of unlimited bandwidth and extended illuminationsources to measure small Doppler shifts of targets external to the apparatus. Monochromatic andpoint-like sources can also be used, thus creating complete flexibility for the illumination source.Although denoted white light velocimetry, the principle can be applied to any wave phenomenon,such as sound, microwaves and light. For the first time, powerful, compact or inexpensive broadbandand extended sources can be used for velocity interferometry, including flash and arc lamps, lightfrom detonations, pulsed lasers, chirped frequency lasers, and lasers operating simultaneously inseveral wavelengths. Superimposing interferometers are created which produce delays independent ofray angle for rays leaving a given pixel of an object, which can be imaged through the interferometer.

I. CLAIMS

(Submitted, not official version)1. A method of imprinting at least one coherent echo on a beam,

comprising:creating an apparent mirror by real imaging or virtual imaging of

an end mirror, wherein the imaged surface of said end mirror definesthe location of said apparent mirror, wherein the imaged center ofcurvature of said end mirror defines the center of curvature of saidapparent mirror;

providing a second mirror optimally having the same curvatureas said apparent mirror; and

superimposing the location of said second mirror with the lo-cation of said apparent mirror in the reflection of a beamsplittingsurface.

2. The method of claim 1, wherein said apparent mirror can beslightly tilted from perfect overlap in order to create a delay timewhich varies with position across the plane of said apparent mirror,wherein said beam can be any wave kind which travels in 2 or 3dimensions.

3. The method of claim 2, wherein said wave kind comprises elec-tromagnetic waves selected from a group consisting of microwaves,visible light, infrared light, ultraviolet light and x-rays.

4. The method of claim 2, wherein said wave kind comprisessound waves.

5. The method of claim 1, wherein said beam is uncollimated,wherein said beam comprises a range of ray angles significantenough to prevent a coherent delay if said beam were to propagatein free space over a distance comparable to the distance betweensaid apparent mirror and said end mirror.

6. The method of claim 1, wherein said real imaging is accom-plished using at least one elements selected from a group consisting


of lenses, curved mirrors, plane mirrors, distributed mirrors andwaveguides.

7. The method of claim 6, wherein said real imaging is accom-plished using at least one transmissive lens and said end mirror.

8. The method of claim 7, wherein said end mirror comprises acurved mirror.

9. The method of claim 6, wherein said end mirror comprises aplane mirror in combination with a lens which together effectivelyact as a curved mirror.

10. The method of claim 6, wherein said real imaging is accom-plished by a transmissive lens, at least one center curved mirrorand said end mirror.

11. The method of claim 10, wherein said end mirror comprisesa curved mirror.

12. The method of claim 10, wherein said end mirror comprisesa plane mirror in combination with a lens which together effectivelyact as a curved mirror.

13. The method of claim 1, wherein the optics forming saidapparent mirror are achromatic so that the location of said ap-parent mirror is approximately constant over a significant range ofwavelengths.

14. The method of claim 1, wherein at least one of the elementsforming said apparent mirror is a distributed mirror comprising aneffective reflective surface depth that is a function of wavelength.

15. The method of claim 14, wherein said effective reflectivesurface depth wavelength dependence compensates for chromaticdispersion of other elements forming said apparent mirror so thatthe net chromatic abberation of said apparent mirror is reduced.

16. The method of claim 1, wherein said virtual imaging isaccomplished using lenses and a curved end mirror.

17. A method of imprinting an infinite series of diminishingamplitude coherent echos on a beam, comprising:

creating an apparent mirror by real or virtual imaging of a firstend mirror, which may be partially transmitting;

providing a second end mirror having the same curvature andcurvature polarity as said apparent mirror, wherein said second endmirror may be partially transparent;

2-2

superimposing said second end mirror with said apparent mirrorso that a recirculating path is created having a magnification ofpositive unity for a roundtrip along the optic axis, wherein saidroundtrip is from said first end mirror to said second end mirrorthen back to said first end mirror;

introducing said beam to said recirculating path by partial trans-mission through a partially reflecting surface; and

extracting a portion of said beam from said recirculating paththrough a partially transmitting surface, wherein the roundtriptravel time along said recirculating path determines the interfer-ometer delay time, wherein said portion comprises an infinite seriesof diminishing amplitude coherent echos.

18. The method of claim 17, wherein said beam is uncolli-mated, wherein said beam comprises a range of ray angles sig-nificant enough to prevent a coherent delay if said beam were topropagate in free space over a distance comparable to the distancebetween said apparent mirror and said first end mirror.

19. The method of claim 17, wherein said real imaging is accom-plished using at least one element selected from a group consistingof lenses, curved mirrors, plane mirrors, distributed mirrors andwaveguides.

20. The method of claim 19, wherein said real imaging is ac-complished by at least one lens and said end mirror.



23. The method of claim 20, wherein said real imaging is accom-plished using at least one curved mirror and and said end mirror.



26. A method of coherently delaying a beam, comprising;creating an apparent mirror by real or virtual imaging of an

end mirror, wherein the imaged surface of said end mirror definesthe location of said apparent mirror, wherein the imaged center ofcurvature of said end mirror defines the center of curvature of saidapparent mirror; and

reflecting said beam off said apparent mirror, wherein the beamis delayed by the interval of round trip travel between said appar-ent mirror and said end mirror and back to said apparent mirror,wherein the beam can be any wave kind which travels in 2 or 3dimensions.

27. The method of claim 26, wherein said wave kind is visiblelight.

28. The method of claim 26, wherein said wave kind com-prises electromagnetic waves selected from a group consisting ofmicrowaves, visible light, infrared light, ultraviolet light and x-rays.

29. The method of claim 26, wherein said wave kind comprisessound waves.

30. The method of claim 26, wherein said wave kind comprisesmatter waves.

31. The method of claim 26, wherein said beam is uncolli-mated, wherein said beam comprises a range of ray angles sig-nificant enough to prevent a coherent delay if said beam were topropagate in free space over a distance comparable to the distancebetween said apparent mirror and said end mirror.

32. The method of claim 26, wherein said real imaging is accom-plished using at least one element selected from a group consistingof lenses, curved mirrors, plane mirrors, distributed mirrors andwaveguides.

33. The method of claim 32, wherein said real imaging is ac-complished with at least one lens and said end mirror.


35. The method of claim 33, wherein said end mirror comprisesa plane mirror in combination with a lens which together effectively

act as a curved mirror.

36. The method of claim 32, wherein the real imaging is accom-plished with a transmissive lens, at least one center curved mirrorand said end mirror.


38. The method of claim 36, comprises a plane mirror in combi-nation with a lens which together effectively act as a curved mirror.

39. The method of claim 26, wherein the optics forming saidapparent mirror are individually or as a group achromatic so thatthe location of said apparent mirror is approximately constant overa significant range of wavelengths.

40. The method of claim 26, wherein at least one of the elementsforming said apparent mirrors comprises a distributed mirror whoseeffective reflective surface depth is a function of wavelength.

41. The method of claim 40, wherein the wavelength dependenceof said effective reflective surface depth compensates for chromaticdispersion of other elements forming said apparent mirror so thatthe net chromatic abberations of said apparent mirror is reduced.

42. The method of claim 26, wherein said virtual imaging isaccomplished by a lens and a curved end mirror.

43. An apparatus for imprinting at least one coherent echo on abeam, comprising:

means for creating an apparent mirror by real imaging or virtualimaging of an end mirror, wherein the imaged surface of said endmirror defines the location of said apparent mirror, wherein theimaged center of curvature of said end mirror defines the center ofcurvature of said apparent mirror;

providing a second mirror optimally having the same curvatureas said apparent mirror; and

superimposing the location of said second mirror with the lo-cation of said apparent mirror in the reflection of a beamsplittingsurface.

44. The apparatus of claim 43, wherein said wave kind compriseslight.

45. The apparatus of claim 43, wherein said wave kind com-prises electromagnetic waves selected from a group consisting ofmicrowaves, visible light, infrared light, ultraviolet light and x-rays.

46. The apparatus of claim 43, wherein said wave kind comprisessound waves.

47. The apparatus of claim 43, wherein said beam is uncolli-mated, which is to say that there is a range of ray angles significantenough to prevent a coherent delay if said beam were to propagatein free space over a distance comparable to the distance betweensaid apparent mirror and said end mirror.

48. The apparatus of claim 43, wherein said real imaging isaccomplished using at least one element selected from a group con-sisting of lenses, curved mirrors, plane mirrors, distributed mirrorsand waveguides.

49. The apparatus of claim 48, wherein said real imaging is ac-complished by one or more transmissive lenses and said end mirror.

50. The apparatus of claim 49, wherein said end mirror com-prises a curved mirror.

51. The apparatus of claim 49, wherein said end mirror com-prises a plane mirror in combination with a lens which togethereffectively act as a curved mirror.

52. The apparatus of claim 48, wherein said real imaging isaccomplished by a transmissive lens, at least one center curvedmirror and said end mirror.



55. The apparatus of 43, wherein the optics forming said appar-ent mirror are individually or as a group achromatic so that thelocation of said apparent mirror is approximately constant over asignificant range of wavelengths.

2-3

56. The apparatus of 43, wherein at least one of the elementsforming said apparent mirror comprises a distributed mirror whoseeffective reflective surface depth is a function of wavelength.

57. The apparatus of 56, wherein the wavelength dependenceof said effective reflective surface depth compensates for chromaticdispersion of other elements forming said apparent mirror so thatthe net chromatic abberations of said apparent mirror are reduced.

58. The apparatus of claim 43, wherein said virtual imaging isaccomplished by one or more lenses and a curved said end mirror.

59. An apparatus for imprinting an infinite series of diminishingamplitude coherent echos on a beam, comprising:

means for creating an apparent mirror by real or virtual imagingof an end mirror, which may be partially transmitting;

a second end mirror having the same curvature and curvaturepolarity as said apparent mirror, wherein this second end mirrormay be partially transparent, wherein said second end mirror is su-perimposed with said apparent mirror so that a recirculating pathis created having a magnification of positive unity for a roundtripalong the optic axis, wherein said roundtrip is from said first endmirror to said second end mirror then back to said first said endmirror;

a partially reflecting/transmitting surface through which saidbeam can be introduced into said recirculating path;

a partially reflecting/transmitting surface through which aportion of said beam from said recirculating path can be ex-tracted, wherein the interferometer delay time is determined bythe roundtrip travel time along said recirculating path.

60. The apparatus of claim 59, wherein said beam is uncolli-mated, wherein said beam comprises a range of ray angles signifi-cant enough to prevent a coherent delay if said beam were to prop-agate in free space over a distance comparable to distance betweensaid apparent mirror and said first end mirror.


62. The apparatus of claim 61, wherein said real imaging isaccomplished by at least one lens and said end mirror.



65. The apparatus of claim 62, wherein said real imaging isaccomplished using one or more curved mirrors and said end mirror.



68. An apparatus for coherently delaying a beam, comprising;an end mirror; andmeans for creating an apparent mirror by real or virtual imaging

of said end mirror, wherein the imaged surface of said end mirrordefines the location of said apparent mirror, wherein the imagedcenter of curvature of said end mirror defines the center of curvatureof said apparent mirror, wherein said beam is reflected off saidapparent mirror, wherein the beam is delayed by the interval ofround trip travel between said apparent mirror and said end mirrorand back to said apparent mirror, wherein the beam can be anywave kind which travels in 2 or 3 dimensions.

69. The apparatus of claim 68, wherein said wave kind comprisesvisible light.

70. The apparatus of claim 68, wherein said wave kind com-prises electromagnetic waves selected from a group consisting ofmicrowaves, visible light, infrared light, ultraviolet light and x-rays.

71. The apparatus of claim 68, wherein said wave kind comprisessound waves.

72. The apparatus of claim 68, wherein said wave kind comprisesmatter waves.

73. The apparatus of claim 68, wherein said beam is uncol-limated, wherein said beam comprises a range of ray angles sig-nificant enough to prevent a coherent delay if said beam were topropagate in free space over a distance comparable to the distancebetween said apparent mirror and said end mirror.


75. The apparatus of claim 74, wherein said real imaging isaccomplished by at least one lens and said end mirror.



78. The apparatus of claim 74, wherein the real imaging is ac-complished by a transmissive lens, at least one center curved mirror,and said end mirror.



81. The apparatus of 68, wherein the optics forming said appar-ent mirror are individually or as a group achromatic so that thelocation of said apparent mirror is approximately constant over asignificant range of wavelengths.

82. The apparatus of 68, wherein at least one of the elementsforming said apparent mirror comprises a distributed mirror whoseeffective reflective surface depth is a function of wavelength.

83. The apparatus of 82, wherein the wavelength dependenceof said effective reflective surface depth compensates for chromaticdispersion of other elements forming said apparent mirror so thatthe net chromatic abberations of said apparent mirror is reduced.

84. The apparatus of claim 68, wherein said virtual imaging isaccomplished by a lens and a curved said end mirror.

85. The method of claim 2, wherein said wave kind compriseslight waves.

86. The method of claim 2, wherein said wave kind comprisesmatter waves.

87. The apparatus of claim 43, wherein said wave kind comprises

matter waves.

Acknowledgments


IL-9864NoisePair15.tex 12/03

Part 3

Noise-Pair Velocity andRange Echo-LocationSystem

US Patent 5,872,628Issued Feb. 16, 1999


An echo-location method for microwaves, sound and light capable of using incoherent and arbitrarywaveforms of wide bandwidth to measure velocity and range (and target size) simultaneously to highresolution. Two interferometers having very long and nearly equal delays are used in series with thetarget interposed. The delays can be longer than the target range of interest. The first interferometerimprints a partial coherence on an initially incoherent source which allows autocorrelation to beperformed on the reflected signal to determine velocity. A coherent cross-correlation subsequent tothe second interferometer with the source determines a velocity discriminated range. Dithering thesecond interferometer identifies portions of the cross-correlation belonging to a target apart fromclutter moving at a different velocity. The velocity discrimination is insensitive to all slowly varyingdistortions in the signal path. Speckle in the image of target and antenna lobing due to parasiticreflections is minimal for an incoherent source. An arbitrary source which varies its spectrumdramatically and randomly from pulse to pulse creates a radar elusive to jamming. Monochromaticsources which jigger in frequency from pulse to pulse or combinations of monochromatic sources cansimulate some benefits of incoherent broadband sources. Clutter which has a symmetrical velocityspectrum will self-cancel for short wavelengths, such as the apparent motion of ground surroundingtarget from a sidelooking airborne antenna.

I. INTRODUCTION

A. Field of the Technology

The present invention relates to the simultaneous mea-surement of target velocity and position by microwaves(radar), sound (sonar and ultrasound) and light, andmore specifically, it relates to the use of long delay in-terferometric techniques, short coherence length illumi-nation and the Doppler effect to measure velocity andrange.

B. Description of the Background

Measuring the velocity and range of an object remotelyby the response of reflected waves (microwaves, light andsound) is an important diagnostic tool in a variety offields in science and engineering. In many cases, the ve-locimetry and range finding are accomplished by oppositekinds of illuminating waves. Traditionally, highly coher-ent and therefore narrowbanded illumination is used for


velocimetry, so that the Doppler frequency shift in the re-flected waves is larger than the illumination bandwidthβ. This allows the Doppler frequencies of moving targetsto be easily distinguished from those of stationary ob-jects by the use of narrowbanded filters. In contrast, forrange finding a short coherence length Λ is desired, sincehalf the coherence length defines the range resolution. Ashort coherence length implies broadband illuminationbecause of the reciprocal relationship Λ ∼ c/β, where cis the speed of the waves. Thus, illumination optimizedfor conventional velocimetry is very different from thatoptimized for range finding.

For example, monochromatic laser illumination is con-ventionally used for optical velocimetry, whereas broad-band and incoherent light such as white light is usedfor rangefinding, which in optics is called optical coher-ence tomography. Pulse Doppler radars are used for ve-locimetry having monochromatic waves enveloped into∼1 µs pulses. The range resolution of these is limitedby the ∼300 meter pulse envelope length. In contrast,range finding or mapping radars use illumination havinga much shorter coherence length. This is often achievedby increasing the bandwidth of the waveform through fre-quency modulation or the repetitive use of a coded pulseto simulate randomness.

A theoretical optimum waveform for many range find-

3-2

Arbitrarysource

Velocity onlyAutocorrelation

~Crosscorrelation

( )2∫dt

8

9

( )2∫dt Range and velocity

Moving referenceframe delay

3

11

5

5Doppler scaling of waves

4

Interferom.1 Partial coherence imprinted

Interferom.2

Partial coherence interpreted

10

2

2

1

7

6

3

12

Figure 1A illustrates a block diagram of the invention.

ing applications is wideband and incoherent, analogousto white light generated by incandescence. For radarand sonar this could be generated by an electronic noisesource. A truly incoherent waveform is random andtherefore never repeats. This is an advantage becauseany speckle or temporary coherences in the echolocationtarget image is blurred away, improving the observationof detail. Incoherent illumination will not suffer the in-terference effect between the transmitted beam and par-asitic reflections from the Earth, thus minimizing thelobed characteristics of antennas in the altitude direc-tion. A further advantage of incoherent illumination overa short pulse or frequency modulated pulse having thesame coherence length is that a random waveform canbe arbitrarily long without repetition. That is, its time-bandwidth product can be arbitrarily large. The longerpulse carries correspondingly more energy to the target,which increases the maximum range of detection.

An important desirable trait of echolocation systemsis the ability to measure velocity and range simultane-ously to high resolution. In radar, the velocity resolutionis used to separate moving targets from ground clutter.Once detected, the range and if possible, the size of thetarget is desired to be known. In medical ultrasound, themotion of the blood can distinguish blood vessels fromsurrounding tissues. A map of the vessels having highspatial resolution and color coded by velocity is the goalof these devices.

Measuring the velocity and range simultaneously tohigh resolution is possible by measuring the range to highresolution at two different, but well defined times and

finding the rate of change. This is accomplished in someradars using short coherence waves and by cross correlat-ing the reflected and transmitted signals. A disadvantagewith a cross-correlation is that distortions in the wavessuffered anywhere between the transmitter and receivercan broaden the cross-correlation peak, and hence canblur the velocity determination. Such distortions couldinclude distortions in the final stages of the transmittingamplifier, the effect of propagation through the atmo-sphere, the target albedo spectrum such as resonances,and reverberations from previous pings.

A different method for determining velocity is the au-tocorrelation. This is a comparison of the received signalwith itself delayed by a time τ2. For those waveformswhich produce an autocorrelation peak, the peak shiftswith velocity v by an amount ∆τ2 = 2τ2v/c. The ad-vantage of the autocorrelation is that it will not broadenby slowly varying distortions, which includes many prac-tical distortions. Thus the autocorrelation will give anaccurate velocity under distortions that would cause across-correlation to blur.

The problem is that autocorrelations cannot be usedwith the most desirable illumination, perfectly randomand ever changing waveforms, because these waveformsare by definition incoherent over all delay scales. Themethod of the present invention is a solution to thisproblem. In addition to velocimetry using incoherent il-lumination, the invention is capable of measuring rangeand velocity simultaneously to high resolution, providinga way to discriminate moving targets from backgroundclutter.

3-3


It is an object of the present invention to use an ar-bitrary waveform, which may be incoherent, to measurethe range and velocity of a target by autocorrelating andcross-correlating the waves reflected from the target.

It is also an object of the present invention to provide amethod to produce partially coherent illumination fromarbitrary waveforms which could be incoherent.

Properties of the target which produce a dilation orcontraction of the illuminating waveform can be mea-sured by this method, where the illuminating beam caneither reflect or pass through the target. For example, ifthe target consists of a region containing plasma, the timerate of change of the plasma density will dilate or contractthe waveform of the reflected or transmitted illuminationsimilar to the Doppler effect of a moving reflector.

A transmitter comprises a wave source, an interferom-eter having a delay τ1, and an antenna or transducer forsending the waves toward the target. The transmittinginterferometer imprints the source with partial coherencefor a very specific delay τ1, and uses this for illuminatingthe target. A receiver comprises an antenna or trans-ducer for detecting the waves reflected from the target,and a second interferometer of delay τ2 nearly identicalto τ1. The output of the receiving interferometer is con-verted to a time-averaged intensity which constitutes anautocorrelation. The autocorrelation can be interpretedto yield target velocity. The output of the receiving in-terferometer is also added to a copy of the source wave-form, copied prior to the transmitting interferometer, anddelayed in time by τ3 to form a combined signal. Thetime-averaged intensity of this combined signal versus τ3

constitutes a cross-correlation which gives range infor-mation. By slightly dithering the value of (τ1 − τ2), theportions of the cross-correlation associated with a targetmoving at a specific velocity will fluctuate synchronouslywith the dither. Thus the range of targets having specificvelocities can be identified in the cross-correlation apartfrom other targets having different velocities. This allowsa moving aircraft to be detected even though its radarreflections are overlapped by reflections from stationaryground clutter.

The transmitting interferometer allows the use of a in-coherent source by imprinting a partial coherence to it,for a very specific delay value of τ1. On other lengthsscales, the illumination sent to the target appears in-coherent. Without the transmitting interferometer, theautocorrelation at the receiver would not produce a peakwhich can be interpreted into velocity, and the cross-correlation would not depend on the dither of (τ1 − τ2),so there would be no velocity discrimination in the rangemeasurement.

By the use of a incoherent wideband source waveform,high range and velocity resolutions are achieved simulta-neously due to the short illumination coherence length.Most of the randomness of the source is preserved in theimprinted waveform used for illumination. This random-

ness blurs any speckle in the cross-correlation due to tem-porary coherences. A random waveform delivers a largeenergy to the target without repeating. Repetition isavoided to reduce range and velocity ambiguities.

It is an object of the interferometers to produce froman applied waveform an output waveform consisting of anundelayed waveform and a finite number of delayed wave-forms, called echos. The time separation between the un-delayed and each of the delayed waveforms is defined theinterferometer delay. The delayed and undelayed wave-forms must be coherent replicas of each other. However,it is allowed that the undelayed and delayed waveformsdiffer from the applied waveform. That is, distortion is al-lowed in the interferometer mechanism provided the dis-tortion affects undelayed and delayed waveforms equally.This allows inexpensive schemes to create the interferom-eter delays which may not treat all fequency componentsequally.


A. Overview

Figure 1A shows an overview of the invention, consist-ing of a means for performing interferometry, autocor-relations and cross-correlations, called the apparatus 3,and a means 5 such as an antenna or transducer for con-verting the waves 1 propagating between the target 4 andapparatus into the waves 2 which will carry the informa-tion through the apparatus. The source 10 generates awaveform which can be nearly arbitrary, the exceptiondescribed later. The preferred embodiment is for a wide-band waveform having a very short coherence length Λand which is incoherent on all other length scales, andwhich is ever evolving in a random way so that specklein the correlations are washed out over time.

The source passes through an interferometer 6, calledthe transmitting interferometer, which has a very longdelay τ1 compared to Λ. A second interferometer, calledthe receiving interferometer 7 has a delay τ2 very nearlymatched to τ1. The time averaged intensity 8 of thereceiving interferometer output is equivalent to an au-tocorrelation. The transmitting interferometer imprintsa partial coherence on the source waveform that allowsthe subsequent autocorrelation of the received signal toproduce a peak at delays τ2 near τ1. The peak’s shiftdetermines the velocity. A cross-correlation 9 of the re-ceiving interferometer output and the source yields rangeand velocity information. By dithering τ2, the portionsof the cross-correlation corresponding to a moving targetof a particular velocity can be distinguished from otherobjects having different velocities. Thus output 9 can becalled a velocity discriminated range signal.

3-4

Delay

1BTwo-path

Interferometer

Pairimpulse response

20

t

Figure 1B illustrates a two-path interferometer concept andits impulse response.

Delay

b

1C

|b|<1

RecirculatingInterferometer

N-tupletimpulse response

21

t

Figure 1C illustrates a recirculating interferometer conceptand its impulse response.

B. About the interferometers

The partial coherence is imprinted on the source wave-form by the transmitting interferometer by creating fromthe original waveform a pair or N-tuplet of waveformreplicas, where N is not infinite, and the replicas areseparated in time by τ1. Without this imprinting, anincoherent source would not create an autocorrelationpeak near τ1, because that is the definition of incoher-ence. The preferred embodiments use either a two-pathinterferometer (Fig. 1B) or a recirculating interferome-ter (Fig. 1C). The two-path interferometer creates twoequal strength pulses in its impulse response separatedby τ . The recirculating interferometer creates an infinitenumber separated by τ having a geometrically decreas-ing strength. The number of pulses having at least 50%of the original amplitude is approximately N . Other in-terferometer impulse responses are permissible, providedthe spacing of output pulses is a uniform τ .

As the recirculating gain coefficient |b| approachesunity, the N approaches infinity. An infinite value forN is not preferred because then it is impossible for theoutput waveform to change. An evolving waveform ispreferred so that speckle (temporary coherences) is re-moved by averaging. On the other hand, increasing Nincreases the robustness of the autocorrelation to statis-

tical noise somewhat. The optimum N may depend onthe noise environment, but is likely to be N=2 to 4.

The term “replica” in the definition of the interfer-ometer should be qualified. The undelayed and delayedcomponents of the interferometer output must be coher-ently identical to each other. However, these are allowedto differ from the waveform applied to the interferometer.This is equivalent to allowing a distortion 322 or 320 inthe interferometer mechanism, as shown in Fig. 1E, pro-vided the distortion affects delayed 321 and undelayed325 portions of the waveform equally.

C. Measuring a Doppler Shift

The process that allows a slight Doppler shift to bemeasured from a short coherence length waveform canbe explained by Figures 2A-E. Fig. 2A shows an opticalembodiment of the invention that measures only the ve-locity and not the range, corresponding to output 8 ofFig. 1A. This apparatus is described in U. S. Patent Ap-plication Serial Number 08/597,082, incorporated hereinby reference. Although the waveform in general can be ofarbitrary length, the explanation is simplest by modelingthe source 21 as a single very short pulse 36 having thesame duration as the coherence time Λ/c. The sourcewaveform passes through a transmitting interferometer22, reflects off a target 24, and passes through a receiv-ing interferometer 23. The interferometer delays τ1 andτ2 are approximately equal, their gross value is denotedτ where τ = τ1 ≈ τ2. For example, in one embodimentof Fig. 2A τ ≈10−8 s and τ1 and τ2 may differ by 10−14

s, which is a million times smaller than τ . The timeaveraged intensity of the receiving interferometer com-plementary outputs 〈I+〉 26 and 〈I−〉 27 are detected bydetectors 38 and 39. The integration time of the detec-tors is longer than the coherence time Λ/c.

Figure 2B shows a hypothetical single source pulse.Fig. 2C shows the pair of identical pulses created by thetransmitting interferometer. Fig. 2D shows the pair af-ter reflection from the target 24 moving with velocity vtoward the apparatus. The pulse separation has changedfrom τ1 to τ1D due to the Doppler effect, where D is theDoppler scaling factor D = (1 − 2v/c). That is, duringthe interval τ1, the target has closed its range by 2vτ1 andhence shortened the arrival time of the second pulse by2vτ1/c. (This assumes the illumination reflects normallyfrom the target. For other angles, the term 2v shouldbe replaced be the time rate of change of the combineddistance between the transmitter to target to receiver.)

The receiving interferometer creates a pulse pair foreach of the two pulses applied to it, creating a total offour pulses, shown in Fig. 2E. The two inner pulses 33,34 overlap if τ1D ≈ τ2 within the coherence length Λ,interfering constructively or destructively depending onthe exact time difference. Suppose the total energy ofthese four pulses are integrated by the detector. The twopulses 32, 35 on the early and late ends contribute con-

3-5

Digital delay

A/DDistortion D/A

1D

f

Tra

nsm

issi

onResult of severe distortion

Distortion322 323

Two-pathInterferometer

325

100%1E

324

326

327 327 327

(f) causes fine comb phase variation

320

321

325

Figure 1D illustrates a digital embodiment of a two-path interferometer. Figure 1E illustrates the transmission spectrum of aninterferometer with severe distortion.

stant energy, independent of the Doppler multiplier. Thetwo inner pulses 33, 34 contribute a fluctuating amountof energy, depending on whether there is constructive ordestructive interference. That is, their contribution de-pends on the Doppler multiplier. These fluctuations arecalled fringes. Since 2 of 4 pulses contribute to the fluc-tuating portion, the fringe visibility is 50%. The targetvelocity is determined from the phase of the fringes.

Since long incoherent waveforms can be modeled as asuperposition of many short pulses of duration Λ/c, thisexplanation also applies to a source of random waves hav-ing arbitrary length. It is not required that the detectorbe so slow that it integrates over all 4 pulses, it is onlynecessary to integrate over the inner two 33 and 34, andthus time average over approximately Λ/c. Thus, afteran initial dwell time of τ2, the receiver becomes sensitiveto Doppler shifts, and can detect them as rapidly as onthe time scale of Λ/c. Practical embodiments howeverwill use a much slower averaging time to reduce statisti-cal fluctuations present in incoherent sources.

The apparatus of Fig. 2A used so-called two path inter-ferometers that create a pair of replica waveforms fromthe source. The method also works for so-called recircu-lating interferometers that generate an N-tuplet of repli-cas.

D. Velocity from Phase Shift

The fringes are only formed for a specific range of de-lays, when |τ1D−τ2| = Λ/c. Thus short coherence length

sources produce the best velocity discrimination. This isrelevant when there is more than one target contributingto the reflected signal. For a single target, any coherencelength can be used. The term clutter denotes an unde-sired target, usually of small velocity such as the ground,that also contributes reflections.

The velocity is determined by the shift in phase of thefringes. During the interferometer delay τ , the targetrange has changed by 2vτ , and this divided by the aver-age wavelength 〈λ〉 yields the number of fringes shiftedfrom the zero velocity point. Thus, velocity is the fringeshift times 〈λ〉/(2τ). For the optical example of Fig. 2Awhen τ=13 ns and 〈λ〉=500 nm, each fringe represents≈20 m/s.

E. Equivalence to an Autocorrelation

The fringe phase can be determined from either 〈I+〉 or〈I−〉 singly, or from the difference 〈I+〉−〈I−〉. The latteris called the pushpull signal 29. The subtraction improvesthe fringe contrast in the output, because non-interferingconstant components are eliminated. The pushpull sig-nal versus delay is equivalent to an autocorrelation signalAC(τ2). This can be seen by considering that if the un-delayed wave is A, and the delayed B, then because theintensity detectors are square law devices, the push-pullsignal is (A+B)2−(A−B)2 = 4AB, which after time av-eraging yields a value proportional to the autocorrelationAC =

∫AB dt.

Thus when a square law detector is used after the in-

3-6

24

Delay2

Delay1

<I->

<I+>

21

26

27

-+

Push-pull29

28

1

After transmitting interferometer 22

36

Before transmitting interferometer 22

2A

1D

2∫ dt

After receiving interferometer 23

32 33

34 35

After target 241D D=1-2v/c

22 23

38

39

30

2B

2C

2D

2E

Figure 2A shows an optical embodiment of the invention for measuring velocity. Figure 2B models the source as a single shortpulse. Figure 2C shows the waveform after the transmitting interferometer. Figure 2D shows the Doppler contracted waveformafter reflection from the target. Figure 2E show the four replica pulses created by the receiving interferometer.

terferometer 23 or 7, it is equivalent to an autocorrelationsignal, as in 8. Note that if the signal to noise ratio isfavorable, the push-pull subtraction is not required andeither single interferometer output 〈I+〉 38 or 〈I−〉 39can determine the shape of AC(τ2), since they differ onlyby a constant.

Figure 3A shows that AC(τ2) is a localized group ofsinusoidal fringes having an envelope width of Λ/c. Thefringes underneath the envelope are sinusoidal and areproportional to

〈I〉 ∝ cos[

2π

〈λ〉 (cτ1 − cτ2) −2π

〈λ〉 (cτ1)(

2v

c

)+ φ0

](1)

where φ0 is some phase constant. Equation 1 dependson both delay difference (τ1 − τ2) and target velocity v.

The envelope 50 and the phase of the fringes underlyingthe envelope shift together with velocity by an amount2vτ/c (52) along the delay axis from the zero velocitypoint, which is τ2 = τ1. The velocity can be calculatedfrom either the envelope shift or the phase shift, sincethey are locked together.

F. Targets Separated from Clutter

Figure 3B shows that the presence of two targets hav-ing different velocities creates two fringe groups 54 and56. One group 54 is due to clutter, presumed at zerovelocity. The two groups can be resolved if their delayseparation is greater than Λ/c. Thus the velocity ambi-guity is Λ/2τ . Hence, short coherence length illumination

3-7

12

AC( 2)

c

2 v/c

52

All

50

2

AC( 2) cluttertarget

5456

1

135

137 139

1333A 3B

AC( ) 2 fixed

58

60/c

Target62

Clutter64AC( )

3C

1

2

66 clutter 68 targetAC( 2, )

+ 2 1v/c

c

1 1

3E

Figure 3A shows the autocorrelation signal for the full bandwidth versus receiving interferometer delay. Figure 3B shows theautocorrelation signal for the full bandwidth for two targets having different velocity. Figure 3C shows the autocorrelationat fixed delay versus frequency. Figure 3D shows the autocorrelation at fixed delay versus frequency for two targets havingdifferent velocity. Figure 3E shows a 2-D map of the maxima in the autocorrelation versus essentially frequency and delay fortwo targets having different velocities.

improves both the velocity and range resolution. As anexample, for the radar embodiment if τ=2 ms, and anaircraft traveling at 45 m/s is to be resolved from sta-tionary clutter, then Λ should be 18 cm. This implies abandwidth β = c/Λ of 1.6 GHz.

If the clutter and target are widely separated, then thepresence of a target at an anticipated velocity can be de-tected by measuring AC at a single delay value. However,it is more useful to measure AC(τ2) over a range of τ2,so that unanticipated target velocities can be detected,and to aid in distinguishing the clutter and target fringegroups. The known symmetry of the envelopes can beused to help distinguish the components having differentvelocities.

Measuring AC(τ2) over a range of τ2 can be done inthe apparatus of Fig. 2A either promptly, or by slowlyscanning τ2. If an interferometer mirror such as 30 inthe receiving interferometer is tilted slightly out of align-ment, then the delay difference τ1 − τ2 can be made to

vary transversely across the output beam. This allowsAC(τ2) to be determined promptly for a single sourcepulse by photographing the output fringe pattern or us-ing some other multi-channel detector. Alternatively, theinterferometer can be aligned to produce the same delayuniformly across its output and a single detecting channelused. Then τ2 can be scanned slowly compared to Λ/cto accumulate AC(τ2) over a range. This latter methodis only appropriate if the target velocity changes slowly.

If the radar/sonar observes a set of targets that areunresolvable in range and possess a distribution of veloc-ities, then if the velocity spread exceeds 1/2 fringe shift,the fringe amplitude for the group will begin to blur away.This is an advantage, since it means that clutter such asweather and the seas which have a distribution of ve-locities can self-cancel, aiding separation of target fromclutter.

3-8

Clutter only

velocity

Target only

Convolution

velocity

Ech

o st

reng

th

/c

Net autocorrelation

Target and clutter

velocity

70 72

74 76

78 79

77

4A

4B

4C

Single velocityimpulse response

Velocity distribution

Figure 4A shows the convolution between a narrow velocity distribution and a single velocity fringe impulse response. Figure4B shows the convolution between a broad velocity distribution and a single velocity fringe impulse response. Figure 4C showsthe combination of a narrow and broad velocity distribution and a single velocity fringe impulse response.

G. Clutter Cancellation

Figures 4A-C illustrate that the process of computingthe net autocorrelation fringe pattern is a convolution,indicated by symbol 72, between a velocity distributionand the fringe pattern 74 from a single target having adistinct velocity. The shape 74 could be called a singlevelocity impulse response. Fig. 4A shows that the veloc-ity distribution for a single target having a well definedvelocity is a spike 70. The convolution between the spikeand 74 is a shape 76 very like 74. Fig. 4B shows thevelocity distribution of hypothetical clutter 78. If thedistribution is approximately symmetric and has a widthwhich is greater than 1/2 the fringe wavelength inside 74,then the convolution will be a very weak set of fringes79, much less than what would be implied by the areaunderneath the velocity distribution curve 78. Fig. 4Cshows that the combination of clutter and a single ve-locity target can produce a fringe pattern 77 where theclutter partially self-cancels and contributes a diminishedamplitude, allowing the target to be identified.

The self-cancellation effect for clutter is enhanced as〈λ〉 → 0. Therefore, as long as the desired target doesnot self-cancel, then it is advantageous to use as short ofaverage wavelength as possible. This is in addition to thebenefit of using short Λ for improved velocity ambiguity.

The self-cancellation effect can be used advantageouslyin a moving antenna platform, such as an airborne plat-form, to lessen the contribution of ground clutter and

Va-Va

Equil-range zone

R

Ra

Vp

Apparent rotation(shear) of groundclutter

Non-blurring clutter footprint

82

84

80

Figure 5A

Figure 5 shows the footprint of ground clutter that will remainafter blurring due to a moving antenna.

3-9

effectively narrow the angular resolution of the radar sim-ilar to the principle in synthetic aperture radar (SAR).Fig. 5 shows that from the point of view of the antennaaircraft 80, which is moving transversely at a velocityVp, the ground on either side of the target 82 is rotating.Only the strip of ground 84 within the range resolution ofthe radar around the target needs to be considered. Therotation generates a symmetrical velocity distribution forthe ground clutter which self-cancels for large azimuthvalues. The magnitude of the apparent velocity due torotation at an azimuth distance Ra is Va = RaVp/R. Forazimuth distances below Ra = 〈λ〉R/(4τVp) the appar-ent rotation does not exceed 1/2 a fringe. Thus there willeffectively be a residue amount of ground clutter repre-sented by the footprint that has an azimuth size of Ra.The angle θ in radians that this subtends from the pointof view of the airborne platform is θ = 〈λ〉/(4τVp). Forexample, if Vp=300 m/s, τ=2 ms and 〈λ〉 =1 cm thenθ=0.24. This is a similar value as that computed from aSAR radar when 2 pulses are considered. That is, in boththe present invention and in SAR, the effective transversesize of the antenna is given by the distance the platformmoves during the observation time of the radar.

In SAR, successive received waveforms are added, af-ter offsetting by the repetition time of the emitted pulsetrain, so that a correlation is accumulated. A changingslight phase shift is usually added to mimic the curvatureof a giant curved antenna along the airborne platform’sline of flight, to effectively focus the synthetic antennaat distances other than infinity. Figures 6A and B helpillustrate how the present invention differs from SAR andother prior art. 1) The purpose of SAR (Fig. 6A) is radarmapping, that is, range finding instead of velocimetry,and thus the arrival time of successive reflections are ref-erenced to the emission time of the transmitted pulse todetermine range. In the present invention (Fig 6B); how-ever, to perform velocimetry there is no reference to theemission time of the transmitted pulse. 2) The presentinvention is able to form correlations 92 from an incoher-ent source by passing the source through a transmittinginterferometer 90. This imprints a partial coherence. Theindividual pulses in the source pulse train 94 in Fig. 6Bare labeled alphabetically to indicate their individualityand that they would have no correlation between them.Because the interferometer delay is set to be approxi-mately the pulse repetition period (PRP), each pulse ofthe transmitted pulse train 96 contains components ofat least two individual pulses from the source train 94(more if a recirculating interferometer is used). Thus,when they are superimposed in the autocorrelation pro-cess 92, the two similar components can correlate. Forexample in 92, “A,B” will partially correlate with “B,C”because of the common B component.

In the prior art however, if an incoherent source is used,there will be no correlation in the receiver at 98. Eventhough pulses “A”, “B” and “C” are added so that theysuperimpose in time, there is no coherent correlation be-tween them because they are random and unrelated. Any

correlation would be of an incoherent kind, that is, re-lated to the envelope of the pulses and not the underlyingphase.

H. Plotting the Autocorrelation vs Wavelength

In the velocimeter apparatus of Fig. 2A, prisms (wave-length dispersive elements) could be placed in betweenthe receiving interferometer 23 and the detectors 26and 27. These would create a spectrum that wouldbe recorded by multi-channel detectors. This spectrum,and therefore the autocorrelation would have fringes, asshown in Fig. 3C and 3D for the case of τ2 = τ1. Equa-tion 1 shows that the fringes plotted versus 1/λ are si-nusoidal, with a spatial “frequency” along the 1/λ axisof [c(τ1 − τ2) − 2τ1v]. If τ2 = τ1, then the target veloc-ity is proportional to this spatial frequency 58. Fig. 3Dshows that the clutter 64 can be distinguished from thetarget 62 by its much lower spatial frequency. The plotis versus the variable 1/λ, which is proportional to wavefrequency and should not be confused with the term spa-tial frequency discussed above.

The autocorrelation can therefore can be plotted ver-sus both τ2 and 1/λ, in a 2-dimensional map as shown inFig. 3E, where AC maxima are displayed as black or grayfringes. The gray 68 fringe network associated with a tar-get can be easily distinguished from the black fringes 66of stationary clutter. The networks are the same shape,just translated in delay. The amount of translation yieldsthe velocity. This flexible method of signal presentationmay be useful for distinguishing target from clutter in anoisy environment. For example, if an enemy is jamminga radar with quasi-monochromatic waves, these can beignored most easily in the AC(λ) presentation. If theenemy is jamming with short pulses, then these can bemost effectively isolated by the AC(τ2) presentation.

I. Sensing Target Albedo

If the complementary outputs 〈I+〉 and 〈I−〉 are addedinstead of subtracted, then the envelope 60 of the AC isobtained. This is the power spectrum of the reflected sig-nal. Thus the target albedo spectrum can be measured,after dividing by the known spectrum of the transmit-ted illumination. Changes in the resonances of the targetalbedo can yield information about the structure and ori-entation of the target, in addition to shape informationthat comes from the range signal, discussed later.

J. Pulse Repetition and Echolocation

Figure 1A shows the transmitting and receiving anten-nas to be separate, to indicate that in general the sourcewaveform can be arbitrarily long. However, because re-flections from distant targets are very weak, a practical

3-10

A B C DSource

Prior art

A B C DTransmitted and received

A

B

CIn correlator

No coherent correlationif A≠B≠C≠D 98

6A

Present invention

A B C DSource

A,B B,C C,D D,ETransmitted and received

A,B

B,C

Interfero-meter

≈PRP

In correlator

PRP

Coherent correlationdespite A≠B≠C≠D

92

9690

94

6B

Figure 6A shows the scheme in prior art for correlating pulses. Figure 6B shows the scheme of the present invention forcorrelating pulses.

Incoherentsource

Interferometer ofdelay 1

PRP

A+B B+C C+D D+E

A B C D

1 ≈ integer x PRP

1 ≈ 3 PRP

A+D B+E C+F D+G

116 118

7B

7A

PW

Enveloper

100

104

114

106

102

108

110 112

Figure 7A shows the enveloping of an incoherent waveform into periodic pulses. Figure 7B shows the effect of jiggering theinterferometer delay relative to the pulse envelope period.

3-11

echolocation system will usually operate in a pulsed modeso that strong and immediate reflections from short rangeclutter do not swamp the distant weak signals, and sothat a time dependent gain can be used on the receivingamplifier. Conventionally, the same antenna is used fortransmission and reception, through the use of a duplexerswitch. Fig. 7A illustrates that the expected best radarand sonar embodiment of this invention is to envelope thesource waveform into broad pulses of duration PW, re-peated in a train with a pulse repetition period PRP. ThePW would be determined by the minimum target rangeand the PRP from the maximum target range. For aradar example, for a minimum and maximum range of 5km and 150 km respectively, the PW and PRP would be33 µs and 1 ms, respectively.

To maintain the approximate pulse envelope, the trans-mitting interferometer delay τ1 should be nearly equal toan integer multiple of the PRP. However, τ1 is not re-quired to be exactly proportional to PRP. Fig. 7A showsthe random source signal 100, the enveloping signal 102,and the signal after the enveloper consisting of individualand uncorrelated pulses of a train 104 labeled alphabeti-cally. When τ1 is nearly a multiple of PRP, then the pulsetrain 108 after the interferometer 106 is a pulse trainwhere each pulse is an additive combination of at leasttwo of the individual pulses of the previous train 104,(more than 2 for a recirculating interferometer, Fig. 1C).This generates the partial coherence that allows autocor-relation. For example, pulses 110 and 112 both containa component of the source pulse “B” 114, and thus therecan be interference in the subsequent receiving interfer-ometer. Fig. 7B shows the transmitted pulse train if τ1 isapproximately 3 times PRP. To prevent the enemy fromrapidly determining τ1, then it is advantageous to ran-domly change τ1 after each source pulse of 104. Sucha random jitter in τ1 relative to PRP is indicated inFig. 7B by the variation in overlap in the edges of thepulses 116, 118. The enemy cannot determine τ1 with-out waiting for the later pulse to arrive. However by thenboth pulses have reflected from the target and useful in-formation about the target range and velocity has beenobtained.

K. Explanation in Frequency Domain

A useful analysis of the invention is in the frequencydomain. Fig. 8A shows two replicas of a incoherent sourcewaveform pulse separated in time by τ , as the wave-form would appear after a two-path transmitting inter-ferometer. Each individual replica 120 has a hypotheticalpower spectrum shown in Fig. 8B, having a bandwidthβ. The two-path interferometer has a transmission spec-trum shown in Fig. 8C. It can be described as a combfilter, since it consists of a periodic array of peaks andvalleys having a periodicity 1/τ . For the particular caseof a two-path interferometer the transmission functionis sinusoidal. A recirculating interferometer generating

f0

comb spacing 1/

Bandwidth

120 120

8A

f0

Pow

er

1228B

f0

Tra

nsm

issi

onP

ower

1248C

8D

Figure 8A shows a pair of identical replicas of a pulsed inco-herent waveform. Figure 8B shows the power spectrum of theincoherent waveform before being paired. Figure 8C showsthe fine comb filter nature of an interferometer. Figure 8Dshows the power spectrum of a pair of identical replicas of apulsed incoherent waveform.

a N-tuplet of replicas would produce a comb filter withthe same comb periodicity, but with thinner and non-sinusoidal comb peaks. The overlap between the singlepulse spectrum 122 and the interferometer spectrum 124is the power spectrum of the pulse pair of Fig. 8A.

The fineness of the comb is what allows detection ofsmall Doppler shifts, much smaller than β, and yet thelarge gross spectrum bandwidth β produces a short co-herence length Λ which allows high range resolution. Forexample, if τ=2 ms and β=1.6 GHz, then the comb spac-ing is 6 million times smaller than the gross bandwidth.

The velocimetry aspect of the invention can be con-sidered to be two nearly matched comb filters in series.(Matched in comb periodicity, but not matched in theconventional usage of that term in radar. The second in-terferometer is not a time-reversal of the first.) The out-put of the receiving interferometer is the intensity thatpasses through overlap of both filters integrated over allthe frequencies of the source. When τ2 = τ1 and v=0,then the power spectra perfectly overlap and large poweris passed. Target motion causes the power spectrum ofthe pulse pair reflected from the target to scale by thefactor D = (1 + 2v/c). This scaling causes large fluctu-ations in the overlap which are manifested as fringes in

3-12

Received spectrum against receiving comb filter

f

Inte

nsity

Tra

nsm

itted

inte

nsity

f

1D = 2

1D = 2 + 180°

f

f

1D ≠ 2

1D ≠ 2 + 180°

f

Targetvelocity nearlymatched

Targetvelocity greatlymismatched

130

132

138

139

9A

9B

9C

9D

9E

Figure 9A shows the receiving comb filter compared againstspectrum of wave from target. Figure 9B shows spectrumpassing through receiving comb filter when receiving delayis set to anticipate target Doppler shift. Figure 9C showsspectrum passing through receiving comb filter when delay isincremented by half a fringe over Figure 9B. Figure 9D showsspectrum passing through receiving comb filter when receivingdelay is very mismatched to anticipated target Doppler shift.Figure 9E shows spectrum passing through receiving combfilter when delay is incremented by half a fringe over Figure9D.

the output power of the receiving interferometer. Thiseffect is analogous to the Moire fringes seen when twomeshes of slightly different pitches are overlapped. Thisallows detection of pitch changes too slight to be observeddirectly in a single mesh.

Figure 9A shows the receiving interferometer spectrum130 and the reflected signal spectrum 132 from a pulsepair when the two periodicities are perfectly matched,when τ2 = τ1D. Fig. 9B is the resulting overlap inten-sity spectrum. The integration of this over frequency isthe detector time averaged signal 〈I+〉 which yields theautocorrelation. This would correspond to point 133 ofFig. 3A. As the velocity changes, or if τ2 changes, the pe-riodicity becomes mis-matched. Fig. 9B shows the over-lap under the slight mismatch equivalent to half a fringe,that is, when c(τ1D− τ2) = 〈λ〉/2. The integrated poweris much reduced and corresponds to point 135 of Fig. 3A.The large fluctuations in power generate the large fringesat the center of the fringe group envelope 50.

If τ2 is set to be greatly mismatched to the target ve-locity, then weak or no fringes are produced. This canbe seen because there is little difference in the overlap

integral when τ2 is dithered by half a fringe. Fig. 9Dshows the overlap when τ2 is greatly mismatched fromτ1D. Fig. 9E shows the overlap when τ2 is a half fringegreater. The areas under the two overlap curves 138 and139 are nearly equal and correspond to the portions 137and 139 of Fig. 3A where the AC does not fluctuate sig-nificantly.

1. Undesired Kind of Illumination

For a single target (no clutter), the invention can oper-ate using source waveforms of any kind of coherence, highor low, and still generate fringes, and the phase of thefringes will yield the correct velocity according to Equa-tion 1. However, one specific kind of waveform is notpreferred for this invention. That is a waveform whereΛ ≈ τ1c and which Λ fluctuates, because in that case thesource will already have a fine comb spacing nearly thesame as the transmitting interferometer, and thus thetransmitted intensity will fluctuate strongly with smallchanges in source Λ. These fluctuations do not preventdetermination of the velocity, because they do not affectthe fringe phase in the receiving interferometer output.However, they may be a nuisance because the signal leveland therefore the fringe amplitude will vary erratically.Because of this, it is recommended that the source shouldnot be a repeating pattern with a repetition period closeto τ1, unless the period is carefully stabilized to be con-stant.

2. Comparison to Matched Filters

The double interferometers of this invention should notbe confused with the matched filters of pulse compressionradar. The filters of the two techniques have differentpurposes and behaviors. Furthermore, the pulse com-pression technique will not detect a Doppler shift of 1part in 106 using the same source bandwidth as this in-vention. In pulse compression, the first filter stretchesthe pulse by a factor of 100 to 1000 to lower the peakpower by this same factor. A single pulse is outputtedfor a single pulse inputted. In the present invention, twoidentical pulses are outputted, not one, and each are notsignificantly broadened from the original. Consequentlythe peak power is only reduced by a factor of 2. In pulsecompression, the second filter is a time reversal of thefirst, so that in the receiver the single broad reflectedpulse is compressed to its original narrow width. In thepresent invention, the second interferometer is not a timereversal of the first, since it does not output a single pulsefor two input pulses.

In pulse compression, the intensity transmission spec-trum does not contain valleys which completely eliminatesome frequencies. This is because spectral componentsare merely moved around in time, not eliminated. With-out these narrow valleys it is not possible for the pulse

3-13

compression scheme to measure a Doppler shift that is 1part in 106 narrower than the source bandwidth.

L. Expected Practice for Radar

The expected practice of a radar embodiment would beto initially use the velocity-only signal 8 (of Fig. 1A), orthis signal with simple range gating, to search for thepresence of a moving target and measure its velocity.Once the velocity is known, the velocity discriminatingcross-correlation signal 9 can be used to determine therange and size of the target. This is done by settingτ2 = τ1D for the expected target velocity, and settingthe moving reference frame delay 11 to move at the ex-pected velocity. Then τ2 is dithered by a slight amount,equivalent to 1/2 of a fringe, to cause the portion of thecross-correlation representing the target to fluctuate insynchronism with the dither.

The cross-correlation signal 9 is formed by adding thereceiving interferometer output 12 to a delayed copy ofthe source waveform, which is delayed by the movingreference frame delay τ3 11. This sum is then squaredand time averaged over duration at least as long as Λ/cby element 9.

1. Moving Reference Frame

The purpose of the moving reference frame delay τ3 isto reduce the motion of the target in the output cross-correlation signal 9 so that slow electronics after the timeintegration at 9 can display it without blurring its posi-tion. Secondly, the delay is set so that if the source wasa short pulse, the copied source waveform correlates withthe inner two pulses 33, 34 and not the outer pulses 32,35 of the output of the receiving interferometer. Thus,the gross value of τ3 is set to τ3 = τ1 + 2R/c, where Ris the expected range. The rate of change of τ3 is set tothe expected target range rate, dτ3/dt = −2v/c.

The cross-correlation aspect of the present inventioncan be explained by considering the output at 12 to bethe source waveform 10 passing through a single filterhaving the same impulse response as the combinationof interferometers 6 and 7 in series. This effective im-pulse response has the form shown in Fig. 10A and 10B.If τ1D = τ2, then the two interferometers are exactlymatched and their combined impulse response has a cen-tral peak 150 formed from the constructive interferenceof the two inner pulses 33, 34. If τ2 is increased by halfa fringe, that is if τ1D = τ2 + λ/2c, then the centralpeak disappears 152 because of destructive interferencebetween the inner two pulses. The outer two lobes 154,156 are independent of the dither. Thus the signal 12to be cross-correlated contains portions identified withthe target that fluctuate synchronously with the ditherapplied to τ2, provided the target velocity matches theanticipated velocity.

When 1D= 2

2 dithered154

150

156 154 156

10A

When 1D≠ 2

~several c

Height constant as 2 dithered158 160

10B

Figure 10A shows net impulse response of the transmitter andreceiver interferometers in series when delays are matched forthe target Doppler shift and one delay is dithered by half afringe. Figure 10B shows net impulse response of the trans-mitter and receiver interferometers in series when delays aregrossly mismatched for the target Doppler shift and one delayis dithered by half a fringe.

For clutter having velocity that does not cause a matchbetween interferometers 6 and 7, the combined impulseresponse is insensitive to dither in τ2. Fig. 10A showsthat in this case the combined impulse response has twopeaks 158, 160 instead of one in the center, and the heightof these does not fluctuate with τ2 dither because theydo not overlap. Thus, the portions of the signal at 12associated with clutter does not vary synchronously withdither.

If there is no distortion anywhere in the signal pathbetween the source and signal 12, then the width of thepeak 150 in the combined impulse response is infinitelynarrow. In this case, for the target having its velocityexactly matched the cross-correlation signature will notbe any wider than it would be without the interferom-eters. The presence of distortions will cause the cross-correlations signatures to blur. However, the velocitydiscrimination will remain sharp with respect to the ve-locity parameter because it depends on autocorrelation.

The cross-correlation signal plotted versus range and τ2

would have the schematic character indicated by Fig. 11.The data could be processed to present only the portionsof the signal fluctuating synchronously with the dither inτ2. The target 170 would stand out from clutter 172 atthe same range because it would have a sinusoidal depen-dence on τ2. The projections 174 of the data integratedalong the range axis would be similar in appearance toAC(τ2) shown in Fig. 3B.

M. Several Embodiments

Figure 12 shows the invention embodied for radar,sonar or ultrasound or any wave which can be transducedinto an electrical signal by the antenna or transducer 180.

3-14

v/c

velocity)

Range

(mov

ing re

fere

nce

fram

e)

Cro

ssco

rrel

atio

nProjections

Target

Stationaryclutter

172170

174 174

Figure 11

Figure 11 shows a 2-dimensional plot of cross-correlation sig-nal versus receiving interferometer delay and the delay corre-sponding to range in the moving reference frame.

A source of electronic noise 182 is enveloped at 184 intopulses of width PW and repetition period PRP, consis-tent with conventional usage of a single transducer andduplexer 187.

1. Ultrasound

For ultrasound, separate transmitting and receivingtransducers can be used effectively. The transducer mustrespond to or actuate the field of the wave, such as pres-sure in the case of sound, and not a time averaged quan-tity such as sound intensity. The transducer must becoherent so that the same waveform always produces thesame response. For example, Fig. 13 shows a possiblemedical embodiment of the transducers. Laser light 200propagating through an optical fiber 202 vaporizes tis-sue inside a medical patient’s body at 204 and generatessound, which reflects off a target 206 and is detected bya transducer 208. This embodiment will work providedthe vaporization is not so intense during the undelayedwaveform component that the tissue responds differentlyfor the delayed waveform component.

2. General Operation

In all the preferred embodiments, simple range gat-ing can be performed in the conventional manner on thereceived signal to reduce clutter signal from range areasknown to not contain the target. This is shown in Fig. 12at 185. There is no advantage to having the range gatewidth less than PW. This range gating is an incoherentprocess, as opposed to the coherent cross-correlation at188, because the gate width is much longer than 〈λ〉.

To form correlations, the signals must pass through anon-linear device 192 that creates a squared componentthat is time averaged over at least Λ/c. This squaredcomponent corresponds to the intensity of the signal.The receiving interferometer 190 is shown duplicated

only for clarity. The outputs 194 and 196 can generatedonce and then used in several spots to form the vari-ous kinds of cross-correlations and autocorrelations. Theprompt formation of the autocorrelation AC(τ2) over arange of τ2 can be accomplished by multiple autocorre-lators 181 having slightly different delays, but an easiermethod is to use Fourier transforms, as in Fig. 14. Afterthe waveforms have been Fourier transformed by 220, thecross-correlation is produced by multiplying one signalat 221 with the complex conjugated of the other 226 andtaking the inverse Fourier transform of the product 222.The autocorrelation is produced by squaring the magni-tude at 223 before taking the inverse Fourier transform.This automatically produces the spectral dependence ofthese correlations as an intermediate step before they areinverse transformed. By masking the undesired spectralcomponents at 224, the AC spectrum is obtained 225.

N. Simultaneous Multiple Banded Operation

Some of the benefits of incoherent illumination can beachieved by sets of simultaneous monochromatic sourcesthat cover roughly evenly but sparsely the bandwidthrange β. This may be a less expensive way to generatesomewhat incoherent illumination. Secondly, this can beused in conjunction with a filter before the receiver tothwart jamming by the enemy, as shown in Fig. 15A. Thesource 230 generates multiple narrowbands whose config-uration changes randomly rapidly, as shown in Fig. 15B.The spectral information is passed to a multiband nar-rowband filter 232, which prevents harmful jamming sig-nals from entering the sensitive receiver. The spectralinformation is delayed at 234 by the same delay as thetransmitting interferometer, (except for differences in ar-rival time due to different target range), so that the filter232 anticipates the correct narrow bands to pass whenthe legitimate signal from the target arrives. This sys-tem can also be used to perform a kind of range gating,since if the spectral content is changed more rapidly thanPRP, the filter 232 can exclude signals from other thanthe desired range.

1. Means for Achieving Long Delays

A means for implementing a long delay high bandwidthinterferometer for electrical signals is shown in Fig. 16.The idea is that different sections of the signal spectrumcan be heterodyned to a set of lower bandwidth chan-nels by mixing with a set of local oscillators. Multipleparallel channels are used that preferably cover the com-plete original spectral bandwidth β contiguously. A less-preferred embodiment uses channels that sparsely coverthe spectrum. The bandwidth requirements of an indi-vidual channel are proportionately less than the originalsignal. For the example of Fig. 16, a 1600 MHz band-width received input signal 340 is split by a spectrome-

3-15

Low coherencesource

Delay 1

Duplexer

Delay 2

( )2

( )2 ∫dt

∫dt<I->

<I+>

-+

Autocorrelation

Enveloper: repetitionperiod PRP,pulsewidth PW

delay

( )2 ∫dt<I->

( )2 ∫dt<I+>

-

+Crosscorrelation

Delay 2

Dither

Bandpass 1

Bandpass 2

Bandpass 3

Bandpass 4

Bandpass 5

( )2 ∫dt

( )2 ∫dt

( )2 ∫dt

( )2 ∫dt

( )2 ∫dt

AC( ) , fixed 2

Spectrometer

Autocorrelator ( 2= a)

Autocorrelator ( 2= b)

Autocorrelator ( 2= c)

Autocorrelator ( 2= d)

Autocorrelator ( 2= e)

Optionalrange-gating

CC( 3, 2)

Moving reference frame compensation

AC( 2)

AC( ) , full

Delay 2

Similarlytreated

194

196

181

183

188

190

190

190

186187180 184182

192

192

192

192

192

Figure 12

185

Figure 12 shows an embodiment for radar, sonar or ultrasound using essentially analog electrical devices.

200

202

201

206

208

Body

Figure 13

Figure 13 shows an embodiment for biomedical ultrasoundwhere light through a fiber generates sound.

ter 342 into 16 channels of 100 MHz bandwidth spacedevery 100 MHz. The spectrometer could be constructedof a collection of bandpass filters. The set of local oscil-lators 344 heterodyne the different bands to a common100 MHz bandwidth. From this point, all the channelsare identical. The autocorrelation or cross-correlation isaccomplished on the individual channels, and then the

combined output signal formed by reheterodyning withthe same set of local oscillators and summing at 346 or348. An analogous process occurs in the transmitter.

The advantage of this scheme is that less expensiveelectronics can be used to implement the delays and ana-log to digital conversions when the bandwidth is low. Theaverage frequency of the local oscillators are not requiredto be exactly locked in a mathematical relation to theother local oscillators, provided the same local oscillatorset is used for the transmitting and receiving interfer-ometers. Their frequency can wander in the long term.Over the delay ∼3τ , the local oscillator must be stable to±1/4th a cycle, so that for a given channel the signal inthe transmitter interferometer encounters the same delaywithin 1/4th fringe as in the receiving interferometer, in-cluding the time traveling to the target and back. Forthe example of a 10 GHz average frequency and delayof 2 ms, this is a stability requirement of 0.01 part permillion per 6 millisecond, which is readily achievable.

If the local oscillator frequencies are not locked in aregular mathematical relationship, or if the bands do notcover the original bandwidth contiguously and evenly,then this is equivalent to a signal distortion 322 or 320 inFig. 1D prior to the analog-to-digital conversion 323 or af-

3-16

Low coherencesource

Delay 1

Duplexer


delay

FFT

CrosscorrelationCC( , 2)

IFFT

Delay 2

Dither

FFT

Optionalrange-gating

A/DD/A

A/D

AutocorrelationAC( )

IFFT

, full

AutocorrelationAC( )

Subset of IFFT

Moving reference frame compensation

220

224

220

222

222

222

221

223

Figure 14

225

2

2

* 226

223

Figure 14 shows an embodiment for radar, sonar or ultrasound using numerical processing to perform correlations.

Multi narrowbandsource

Delay 1

Duplexer


Spectral bandinformation

Multi narrowbandfilter

Change bands randomlyevery PRP

To cross-correlatorsection

To auto-correlatorsection

Delay 1

234

Spectral passbandinformation

232

230

15A

Figure 15A shows a scheme for using a randomly changing multiband narrowband source and filter to defeat enemy jamming.

ter the digital-to-analog conversion 324. This could pro-duce a hypothetical distorted interferometer spectrumFig. 1E which contains stop bands 327. Since this dis-tortion is applied equally to delayed 321 and undelayed325 signal paths, this distortion will not prevent the for-mation of a useful autocorrelation.

The presence of a dispersion in the time delay, thatis, a frequency dependence to τ , will cause the phase ofthe fine fringe comb to be incoherent from pass band 325to pass band 326. However, this will not blur (disperse)the full bandwidth autocorrelation if the delay dispersioncharacteristics of the transmitting and receiving interfer-ometers are the same. If they are not exactly the same,any residual dispersion in the autocorrelation can be cor-rected mathematically after the time averaging has beenperformed (where it is less computationally expensive),since it corresponds to a systematic tilting or curving of

the network of fringes in AC(λ, τ2) shown in Fig. 3E.

A long delay interferometer can be constructed of a sin-gle mode optical fiber or an acoustical delay line. If theother portions of the apparatus use electrical signals, theelectrical signal modulates an optical beam or a soundwave which travels through the fiber or acoustical delayand is demodulated at the end of the delay back into anelectrical signal. The undelayed signal could also be con-verted to and from light or sound so that the same distor-tions are imparted to it by the conversion process. Thisway the delayed and undelayed paths have the same dis-tortion. The dispersion in the delay should be matchedbetween transmitting and receiving interferometers.

3-17

f

Pow

er

Randomly varying multispectral source

15B

Figure 15B shows an example sequence of spectra for thescheme of Figure 15A.

Sum

mat

ion

MixerShift register

delay 1+

+

+

LO1

0-0.1 GHz

10 GHz

Mixer0-0.1 GHz 10.2 GHz

Mixer0-0.1 GHz

10.1 GHz

Random num.gen. #3

D/A

Random num.gen. #1

Random num.gen. #2

LO2

LO3

D/A

D/A

Net output10 - 11.6GHzincoherent

10-10.1 GHz

10.1-10.2 GHz

Similarly for all 16 channels

analog signalsdigital signals

Source and transmitting interferometer

10.2-10.3 GHz

Individually ~ 100 MHz bandwidth 1600 MHz bandwidth

Shift registerdelay 1


348

Figure 16A

Figure 16 shows a means for creating a high bandwidth longdelay interferometer using parallel heterodyning. (A) showsthe transmitting portion.

Spe

ctro

met

er

Add

er Autocor-relatoroutput

+ ( )2 ∫dt


( )2 ∫dt

+

A/D

Similarly for all channels

Receiving autocorrelator

10 - 11.6GHz input

10 GHz

Mixer0-0.1 GHz

LO1

10.1 GHz

Mixer0-0.1 GHz

LO2

10.2 GHz

Mixer0-0.1 GHz

LO3

10-10.1 GHz

10.1-10.2 GHz

10.2-10.3 GHz

+ ( )2 ∫dt


( )2 ∫dt

+

A/D

+ ( )2 ∫dt


( )2 ∫dt

+

A/D

340 342

344

344

344

346

Figure 16B

Figure 16B shows the receiving portion for a means for creat-ing a high bandwidth long delay interferometer using parallelheterodyning.

O. Optical Embodiments

An optical embodiment for invention which measuresrange while discriminating velocity is shown in Fig. 17.The optical embodiment for the velocity-only signal hasalready been discussed in Fig. 2. The interferometers240 and 242 are constructed similarly and are imagesuperimposing interferometers. The image superimpos-ing condition is required of any interferometer in thisinvention working with an image or using an unparal-lel rays, including interferometers using sound or mi-crowaves that propagate in 3-dimensions as opposed tobeing converted to 1-dimensional electronic signals or 1-dimensional optical signals along a single mode fiber. (Itis possible to construct lenses, mirrors and beamsplittersand thus interferometers analogous to light for sound andmicrowaves.) Practical sources of incoherent broadbandlight emit light over a range of angles or emit over anarea. To select only exactly parallel rays by use of aninfinitely small pinhole at a distance would in the math-ematical limit of zero aperture diameter produce zeropower. Thus all velocimeters intending to use practicalwhite light sources must satisfy the superimposing con-dition, otherwise vanishingly little power can be utilizedfrom them.

The image superimposing condition states that foreach of a plurality of output rays generated by a giveninput ray, the ray position and angle must superimpose.That is, the multiple images created by an interferome-ter must superimpose longitudinally, transversely, and inmagnitude. This creates an interferometer delay whichis independent of ray angle passing through each pixelimage and is preferred to be achromatic. When this con-dition is satisfied, the 3-dimensional interferometer be-haves as a 1-dimensional interferometer, and the detailsconcerning the imaging can be separated from the de-tails concerning the temporal and spectral behavior ofthe velocimetry and range finding.

Embodiments of superimposing interferometers forlight are shown in Fig. 18A-C. More embodiments of su-perimposing interferometers and a more detailed mathe-matical description of the temporal and spectral behaviorof the velocimetry aspect of the invention are describedin U. S. Patent Application Serial Number 08/597,082.Fig. 18A shows an example of an interferometer of theprior art, called a Fabry-Perot, which does not satisfy theimage superimposing requirement and therefore cannotbe used with broadband sources that emit non-parallellight in this invention. The input ray 280 bounces be-tween two partially reflecting mirrors 288 to produce sev-eral output rays 282, 284, 286 which are displaced trans-lationally from each other, not superimposing.

Figure 18B shows a recirculating interferometer whichhas the same temporal properties of the Fabry-Perot, butsatisfies the image superimposing condition. The lens 272images the surface of partially reflecting spherical mirror274 to the other partially reflecting spherical mirror 276with positive unity magnification. The lenses 278 focus

3-18

2

<I-> Camera

<I+> Cam

era

Interf.

1Whitex1

x2≈x1

pzt

Interf.

Delay

3

-+

Cross-correlation

CC(x, 2)

Slightlymisaligned Delay varies across beam

Moving reference framecompensation

252

256

258

240

260

262246

248

254

264

266

242

244

250

Figure 17

Figure 17 shows an open-beam optical embodiment of the invention.

280278 276 272 274 278

271 273 275

18B

280

288288 284 286282Prior art18A

270

268 266

260 262264

18C

Figure 18A illustrates the non-superimposing nature of aFabry-Perot interferometer. Figure 18B illustrates an opti-cal recirculating interferometer satisfying the superimposingcondition. Figure 18C illustrates an optical two-path inter-ferometer satisfying the superimposing condition.

approximately to the center of lens 272 so that collimatedlight applied to the system emerges as collimated. Mir-ror reflectivity of 66% would produce an approximate3-tuplet impulse response, similar to 21 of Fig. 1C Foran input pulse 280, the output rays 271, 273, 275 super-

impose in position and angle.

Figure 18C shows a two-path (called a Michelson) in-terferometer that satisfies the image superimposing con-dition. Lenses 260 and 262 form a relay system whichimages the spherical mirror 264 to the dashed plane 266.The dashed plane superimposes with the image of theplane mirror 268 of the other arm seen in the 50% beam-splitter 270. This will produce a pair impulse responseshown at 20 in Fig. 1B.

In Fig. 17, the PZT 244 dithers the delay τ2 by half afringe. The compensating delay τ3 (246) approximatelyequals τ1 so that a short pulse from the source will inter-fere with the inner pulses 33, 34 of Fig. 2E. The mirror248 position and velocity compensates for the positionand velocity of the target 250 so that path length frombeamsplitter 252 to beamsplitter 254 for the inner twopulses 33, 34 changes little. Lenses in the delay 246 su-perimpose the image of the beamsplitter 252 to the beam-splitter 254. The two paths 256 and 258 emerging frombeamsplitter 252 and recombining at beamsplitter 254must form a superimposing two-path (Michelson) inter-ferometer. That is, lenses 260 and 262 must cause an im-age located in the region near 252 to superimpose in theregion near 254. The beamsplitter 254 is optionally mis-aligned slightly so that the cross-correlation delay variesacross the beam. By recording the image of the beam bymulti-channel cameras 264 and 266 the cross-correlationover a range of delays can be recorded promptly. A push-pull signal obtained by subtracting the complementary

3-19

outputs yields the cross-correlation.Figure 19 shows an embodiment using single-mode op-

tical fibers which is analogous to Fig. 17. The delays areaccomplished by differing lengths of fibers, such as 291.The superimposing condition is moot because images arenot propagated along the single-mode fiber. The beam-splitters of Fig. 17 are replaced by fiber-splitters 290,except for the final beamsplitter 254. This remains anopen-beam beamsplitter so that it can be slightly mis-aligned and cause a delay between beams 296 and 298which varies across the image at 293. This allows thecross-correlation to be recorded over a range of delayspromptly. Lenses 292 at the fiber ends couple between3-dimensional waves interacting with the target 294 to 1-dimensional signals in the fibers. The delay τ2 is ditheredby some means such as straining the fiber spool so thatthe length changes by 〈λ〉/2.

Acknowledgments


3-20

x1

2Interf.

<I-> Camera

<I+> Cam

era

-+

Cross-correlation

CC(x, 2)

1Interf.

White

x2≈x1Moving reference framecompensation

Dither 23

Slightlymisaligned

Delay varies across beam

291

290 292

292

294

291

293296

298

291

Figure 19

Figure 19 shows a fiber-optical embodiment of the invention.

3-21

IV. CLAIMS

(Submitted, not official version)

1. An apparatus, comprising:a source of arbitrary waves;means for imprinting a finite number of coherent echos onto said

arbitrary waves to produce imprinted waves;means for propagating said imprinted waves onto a target,

wherein said imprinted waves are either transmitted through orreflected from said target to produce reflected/transmitted waves;

means for imprinting a finite number of coherent echos onto saidreflected/transmitted waves to produce a combined signal; and

means for time averaging the square of said combined signal tocalculate the velocity of said target.

2. The apparatus of claim 1, further comprising means for pro-ducing a velocity discriminated range value comprising means forcross-correlating said combined signal with said wide bandwidthwaves from said source before said imprinted waves are produced.

3. The apparatus of claim 1, wherein said arbitrary waves com-prise any waveform.

4. The apparatus of claim 1, wherein said arbitrary waves com-prise wide bandwidth waves.

5. The apparatus of claim 1, wherein said arbitrary waves com-prise waves which over the illumination time of the target behaveincoherently or randomly.

6. The apparatus of claim 1, wherein said arbitrary waves com-prise sets of narrowband waves where the maximum and minimumfrequency bands are widely separated.

7. The apparatus of claim 1, wherein said arbitrary waves com-prise

narrowband waves whose instantaneous frequency changes sig-nificantly with time, continuously or discontinuously (chirpedwaves).

8. The apparatus of claim 1, wherein said arbitrary waves com-prise waves whose smallest coherence time is comparable to or lessthan the interferometer delay time.

9. The apparatus of claim 1, wherein said arbitrary waves com-prise waves whose smallest coherence length is comparable to orless than the change in round trip distance between target andapparatus during the interferometer delay time.

10. The apparatus of claim 1, wherein said arbitrary wavescomprise waves selected from a group consisting of waves whosesmallest coherence time is comparable to or less than the interfer-ometer delay time and waves whose smallest coherence length iscomparable to or less than the change in round trip distance be-tween target and apparatus during the interferometer delay time,wherein said waves are non-repeating waves.

11. The apparatus of claim 1, wherein said arbitrary wavescomprise waves selected from a group consisting of waves whosesmallest coherence time is comparable to or less than the interfer-ometer delay time and waves whose smallest coherence length iscomparable to or less than the change in round trip distance be-tween target and apparatus during the interferometer delay time,wherein said waves are repeating waves.

12. The apparatus of claim 1, wherein said arbitrary wavescomprise waves selected from a group consisting of waves whosesmallest coherence time is comparable to or less than the interfer-ometer delay time and waves whose smallest coherence length iscomparable to or less than the change in round trip distance be-tween target and apparatus during the interferometer delay time,wherin said waves are produced by a noise source.

13. The apparatus of claim 1, wherein said arbitrary waves com-prise waves selected from a group consisting of waves whose small-est coherence time is comparable to or less than the interferometerdelay time and waves whose smallest coherence length is compara-ble to or less than the change in round trip distance between targetand apparatus during the interferometer delay time, wherein saidwaves are produced by a digital algorithm or read from a memorydevice.

14. The apparatus of claim 1, wherein said means for imprint-ing a finite number of coherent echos onto said arbitrary sourceof waves to produce imprinted waves comprises delayed and unde-layed components that are coherently identical, wherein all com-ponents are separated by a delay t1, wherein said delayed and un-delayed components can differ from said waves from said source,wherein slight dispersion of the delay with wavelength is toleratedif matched by the same dispersion in the imprinting process ofthe reflected/transmitted waves, wherein there could be one de-layed component and one undelayed component, and wherein therecould be one undelayed component and many delayed componentsof gradually decreasing intensities for increasing time.

15. The apparatus of claim 1, wherein said means for imprintinga finite number of coherent echos onto said arbitrary source ofwaves to produce imprinted waves, wherein said coherent echoscomprise a wave kind in the imprinting process that is any wavekind for which constructive or destructive interference occurs in thecombined intensity in an interferometer when a delayed replica iscombined to the undelayed wave, and the said wave kind in theimprinting process can be 1-, 2- or 3-dimensional, and whereinif said wave kind is 2- or 3- dimensional, the imprinting processmust satisfy the superposition condition where each output rayassociated with an input ray must superimpose in position andangle.

16. The apparatus of claim 1, wherein said means for imprint-ing a finite number of coherent echos onto said arbitrary sourceof waves to produce imprinted waves, wherein said coherent echoscomprise a wave kind in the imprinting process that can be anelectrical signal, a numerical value processed by digital circuits orby mathematical techniques such as Fourier transforms, or repre-sented by modulating another kind of wave such as the intensity,color or polarization of an optical wave.

17. The apparatus of claim 1, wherein said means for imprintinga finite number of coherent echos onto said arbitrary source of wavesto produce imprinted waves, wherein said coherent echos comprisea wave kind in the imprinting process that can be heterodyned to adifferent average frequency, called the intermediate frequency signal(IF), wherein said wave kind can be split into several componentswhich are individually heterodyned to a set of parallel IF signalswhich are individually imprinted and then recombined by a reverseheterodyning process.

18. The apparatus of claim 1, wherein said imprinted waves canbe converted to another wave kind by an antenna or transducer,wherein said wave kind may be electromagnetic waves such as light,microwaves, infrared waves, or vibrations of pressure or strain suchas sound, and could be 1-, 2- or 3-dimensional, and wherein saidwave kind may be any kind for which the target activity being mea-sured produces a dilation or contraction of the waveform returnedto the apparatus.

19. The apparatus of claim 1, wherein said reflected/transmittedwaves can be converted by an antenna or transducer to a secondwave kind before imprinting, wherein said second wave kind com-prises delayed and undelayed components that are coherently iden-tical, wherein all components are separated by a delay t1, whereinsaid delayed and undelayed components can differ from said wavesfrom said source, wherein slight dispersion of the delay with wave-length is tolerated if matched by the same dispersion in the im-printing process of the reflected/transmitted waves, wherein therecould be one delayed component and one undelayed component,and wherein there could be one undelayed component and manydelayed components of gradually decreasing intensities for increas-ing time.

20. The apparatus of claim 1, wherein for 2- and 3-dimensionalwaves, imprinted delay can be a function of position across theimage.

21. The apparatus of claim 1, further comprising a wavelengthdispersive device to organize the channels detecting the output in-tensity versus wavelength.

22. A method, comprising:

3-22

imprinting a finite number of coherent echos onto a wide band-width source of waves to produce imprinted waves;

propagating said imprinted waves onto a target, wherein saidimprinted waves are either transmitted through or reflected fromsaid target to produce reflected/transmitted waves;

imprinting a finite number of coherent echos onto said re-flected/transmitted waves to produce a combined signal; and

time averaging the square of said combined signal to calculatethe velocity of said target.

23. The method of claim 22, further comprising producing avelocity discriminated range value further comprising the step ofcross-correlating said combined signal with said wide bandwidthwaves from said source before said imprinted waves are produced.

IL-9998MultiBeat17.tex 12/03

Part 4

MultichannelHeterodyning forWidebandInterferometry,Correlation and SignalProcessing

US Patent 5,943,132Issued Aug. 24, 1999


A method of signal processing a high bandwidth signal by coherently subdividing it into manynarrow bandwidth channels which are individually processed at lower frequencies in a parallel man-ner. Autocorrelation and correlations can be performed using reference frequencies which may driftslowly with time, reducing cost of device. Coordinated adjustment of channel phases alters temporaland spectral behavior of net signal process more precisely than a channel used individually. Thisis a method of implementing precision long coherent delays, interferometers, and filters for highbandwidth optical or microwave signals using low bandwidth electronics. High bandwidth signalscan be recorded, mathematically manipulated, and synthesized.

I. INTRODUCTION


The present invention relates to interferometry, corre-lation and autocorrelation of wide bandwidth signals, andmore specifically the coherent conversion of wide band-width signals to a set of many lower frequency signalsand subsequent parallel signal processing.


Electromagnetic waves such as microwaves and light,and ultrasound waves are widely used for echolocation,velocimetry and imaging. These techniques involve signalprocessing such as correlation, interferometry, time de-laying, filtering, recording and waveform synthesis. Cur-rently, many of these applications use fairly monochro-matic waves having a long coherence length. If insteadwide bandwidth waves are used, an increased precisionof the location and velocity of the target results due tothe illumination’s shorter coherence length.


U.S. Patent No. 5,642,194, titled “White Light Veloc-ity Interferometer” and U.S. Patent Application SerialNo. 08/720,343, titled “Noise Pair Velocity and RangeEcho-Location System”, both incorporated herein by ref-erence, describe how two interferometers nearly matchedin delay can be used with wide bandwidth illuminationto measure target velocity using processes of interferom-etry and autocorrelation (Fig. 2A). The wideband illu-mination can be used to measure target range using theprocess of correlation (Fig. 2B). These processes of inter-ferometry, correlation and autocorrelation require coher-ent delays. For microwave radars, the delays are chosento be several milliseconds to match radar pulse repeti-tion rates, which in turn are set by the desired maximumtarget range of hundreds of kilometers. Several millisec-onds is a long delay compared to the period of the wave,which could be 30 picoseconds. Electronics, and partic-ularly digital electronics, is the most attractive methodof creating long delays, as well as for performing generalsignal processing. The challenge is that the bandwidthsfound in microwave and optical signals can greatly exceedthe capability of easily available electronics.

For example, suppose we wish to construct a de-vice using 10-30 GHz microwaves having a bandwidthof 20 GHz, using interferometers of 2 ms delay. (Thiscould provide meter/sec velocity measurement and ∼1cm range resolution.) It is impractical to create such a

4-2

Spe

ctro

met

er

Widebandinput signal

+Widebandoutput

DownMixer

LO112

fLO1

Channelsignal

process

UpMixer

f1 (f1-fLO1)=fIF fIF (fIF+fLO1)=f1

10

3

14

16

11 Quadrature component

∆ 1

IF*IF

Mul t ihe te rodyn ing S igna l Processor

Processing at intermediatefrequency, phase shift to

alter spectrum slightlyHeterodyning todown-shift frequency

Inverse-heterodyning toup-shift frequency

Figure 1

Similarly for all channels

41

2

5

6

8

18

20DownMixer

LO2 fLO2

UpMixer

f2 fIF fIF f2

∆ 2

IF*IF

22

Gain norm. 17

Channelsignal

process

7

Figure 1 is a schematic of multiheterodyning signal processor.

+

delay

"A" Interferometer

Widebandsource

+

delay

"B" Interferometer

∫dt2

Time-averagedpower detection

Autocorrelationshift yields targetvelocity

A) 100102

104

115

106108

110112

Autocorrelator

Figure 2

Figure 2A shows a double interferometer velocimeter usingwideband illumination.

long delay by analog cable, because the cable length re-quired is hundreds of km, and serious attenuation anddispersion of the signal would result, in addition to theimpractical weight and cost of the cable. A digital delayline consisting of an analog-digital converter (A/D), shiftregister, and digital-analog converter (D/A) can easilycreate a 2 millisecond delay. However, the 20 GHz inputsignal bandwidth greatly exceeds the bandwidth capabil-ity of commonly available digital electronics, which maybe closer to 0.2 GHz.

The bandwidth of optical signals can be even muchhigher than microwaves and thus even further beyond thecapability of current electronics. (Optical velocimeters

Widebandsource

+ ∫dt2

Correlator

delay Correlation shiftyields targetrange

B)

"A"

"B"

Figure 2B shows the use of correlation and wideband illumi-nation to find the range of a target.

having long delays are useful for measuring extremelyslow velocities.) In spite of lower weight of optical fiber,a spool of fiber hundreds of km long is still large andexpensive, and the attenuation and dispersion throughsuch a length would force the use of repeaters, furtherincreasing cost and weight.

In addition to velocimetry, applications for correlat-ing wide bandwidth optical signals abound in astronomy,for creating long baseline optical telescopes analogous toradio telescope arrays. These could increase the angu-lar resolution by orders of magnitude over current opti-cal telescopes. The challenge here is to correlate two ormore optical signals received many hundreds of kilome-ters apart. Propagating a weak optical signal throughlong optical fibers brings severe attenuation and disper-sion, losing the phase information required for correla-tions. Digitally encoding the time varying optical field

4-3

C)

Wideband signalfrom target

Correlation shiftyields targetangle

Wideband signalfrom target

A greatdistance

+ ∫dt2

Correlator

delay "A"

"B"

Figure 2C shows the use of correlation to find angular positionof a wideband emitting target.

(amplitude and phase) was not feasible previously.A high bandwidth requirement can also come from the

use of multiple input data streams which need to be cor-related to form a 2-dimensional image. Such is the caseof an ultrasound imaging device using multiple detectors(transducers) and using wideband illumination (to reducespeckle). While a single detector produces a 1 to 10 MHzbandwidth signal, which is low enough to be processedby a single channel of electronics, a matrix of 100 suchdetectors requires calculating a large number of correla-tion combinations. This can exceed real-time processingcapacity.

Figures 2A, 2B, and 2C show applications for inter-ferometry, autocorrelation and correlation. Figure 2Ashows a double interferometer velocimeter, which is alsocalled a “white light” velocimeter for optical waves (seeU.S. Patent No. 5,642,194) and a “noise pair” radar formicrowave waves (See U.S. Patent Application Serial No.08/720,343). Waves from a wideband source 100 passthrough an interferometer 102 labeled “A” having a de-lay τA. This imprints a coherent echo on the source toform an illumination beam 104, which is sent to a target115. The presence of the coherent echo causes a comb-filter shape to the power spectrum of waves 104. Thewaves 106 reflected from the target are Doppler shiftedby the target velocity, which dilates the comb-filter spec-trum slightly. The reflected waves pass through a secondinterferometer 108 labeled “B” having a delay τB andwhich imposes its own comb-filter pass spectrum. Thiscreates a detected wave 110. The time-averaged power112 of the detected wave is measured versus delay τB toform an autocorrelation. The autocorrelation will have apeak for delays τB ≈ τA. This is because the maximumamount of power will pass through the “B” interferometerwhen its spectrum matches the Doppler shifted spectrumof the “A” interferometer. Let the position of the peakbe τp. For target velocity v, and for a target reflectinglight 180 back toward the apparatus, the autocorrelationpeak shift is proportional to velocity, τp − τA = (2v/c),where c is the wave speed. Thus very long delay timesproduce sensitive velocity detection. The advantage of

high bandwidth for the source 100 is a smaller coherencelength, which makes the autocorrelation peak narrower,reducing velocity ambiguity. This allows resolving veloc-ities of multiple targets.

An interferometer is a device that coherently sumsan applied signal with a delayed replica. It is a fil-ter with a power transmission spectrum which is si-nusoidal, as shown in Fig. 3A, ideally given by T =(1/2)[1 + cos (ωτ + const)], where “const” is a constant,ω = 2πf is the angular frequency, and τ the interferom-eter delay. This kind of spectrum is also called a “comb-filter”. Figure 3B shows a close-up of the sinusoidal peakshaving a periodicity 1/τ in frequency units and 360 inphase units. (The periodicity of the sinusoids in theseFigures is exaggerated for clarity. For example, a 2 msdelay interferometer would have 2 ms × 20 GHz = 40million peaks from 10 GHz to 30 GHz.)

Note that it is the structure of the comb-filter on thescale 1/τ which produces the velocity measuring effect.Spectral changes much broader and much narrower than1/τ do not significantly affect the phase of the fringes.Thus a less-than-ideal interferometer that broadly modi-fies the spectrum of the applied signal can still be used, aswell as one having slightly non-uniform spacing (phase)of comb peaks, provided both “A” and “B” interferome-ters share the same non-uniformity.

Figure 2B shows an application of high bandwidth sig-nals to find the range of a target, and Fig. 2C to measurethe angle of a target by the difference in arrival timesof a wave at two antenna spaced well apart (such as inradio astronomy). These applications both create a cor-relation between two signals “A” and “B”. The position(in delay-space) of the correlation peak yields the rangeor angle information. Again, the advantage of using ahigh bandwidth source is to create a narrow correlationpeak, which reduces ambiguity of the measurement.

In addition to correlations and interferometry, othersignal processes which benefit from high bandwidth arethe recording and synthesis of waveforms. The time res-olution of these processes is given by the reciprocal ofthe bandwidth, so high bandwidth allows for the short-est time resolution.


It is an object of the present invention to provide tech-niques for coherently combining a set of low bandwidthsignal processing channels to form a collective high band-width signal processing device.

A high bandwidth reference signal, which in many em-bodiments is provided by a set of oscillators convention-ally called local oscillators (LO), but could also be a sin-gle mode-locked signal having many harmonics, providesthe synchronization that preserves the coherence betweenchannels. The phase relationship between the multiplechannels is set to precisely define the temporal responseof the net device. In this way, for the cases of interferom-

4-4

Trans-mission

Frequency0

A)

∆ dil

f1 f2 f3 f4

For delay

- - - - For delay ( ∆

B)

C)

D)

E)

0

∆ dil

∆ dil ∆ dil

Interferometer comb-filter

1/In frequency:

In phase: 360°

Figure 3

Figure 3A shows the comb-filter transmission spectrum of a phase-linear interferometer. Figure 3B shows a detail of theinterferometer transmission spectrum having sinusoidal peaks of period 1/τ . Figures 3C through 3E show that the effect ofslightly changing the interferometer delay is to locally shift the spectrum by an amount proportional to frequency.

eter, filter, autocorrelator, correlator, or delay-line signalprocesses which involve a characteristic delay time τ , theeffective value of τ can be controlled with much higherprecision than by a single channel used alone.

The method of the invention could be called “multi-channel heterodyning”, or “multiheterodyning”, and canbe used to interfere, correlate, autocorrelate, delay, filter,record and synthesize waveforms, as well as combinationsof those processes. The philosophy is to subdivide the in-put signal spectrum into many narrow bands (channels)which are shifted to a lower frequency (called an inter-mediate frequency, (IF)) by heterodyning against a set ofreference frequencies (fLO), one per channel. Both thephase and amplitude information of the original inputfrequency components can be preserved in the hetero-dyning. The overall signal process desired for the inputsignal is applied individually to each channel. The chan-nel spectra are spectrally re-assembled by a second het-erodyning process which shifts the frequencies upwardto form a net high bandwidth output signal, or in thecase of correlation and autocorrelation processes the nettime-averaged power is computed and summed over allchannels, omitting the up-heterodyning step. In autocor-relation, correlation, interferometry, or delay, each chan-nel process spectra (Fig. 4A) is similar for all channels.Then the net spectrum (Fig. 4B) comprises a set of these

200, 201, 202, translated upward in frequency via het-erodyning to fill the input bandwidth. The channel spec-tra could also be individualized to create a non-periodicspectrum (Fig. 4C).

Since the phase and amplitude of the input fre-quency components can be determined through thedown-heterodyning process, the entire Fourier spectrumof the input signal (real and imaginary parts across thefull input bandwidth) can be recorded, by recording eachindividual channel and re-assembling the spectral infor-mation at a later time. (Optimal recording desires thechannel bandshapes to contiguously fill the input band-width shoulder-to-shoulder). Knowledge of the full-bandFourier spectrum is a complete description of the inputsignal.

For real-time processes, any arbitrary filtering processthat can process each frequency component independentof the others can be implemented by the invention. (Anexample of a process that is inappropriate is a non-linearprocess where widely separated high and low frequencycomponents are expected to mix and form sum or dif-ference frequency output terms. Such pairs are not easyto form in real-time for the invention because the highand low input frequencies are segregated into separatechannels.) However, the combination of recording an in-put signal, manipulating its recorded Fourier spectrum

4-5

Trans-mission

Frequency00

Effective spectrum for a multiheterodyned interferometer

Trans-mission

Frequency0 ch

0

fLO1 fLO2 fLO3 fLO4 fLO5 etc.

Spectrum for a channel when all channels are similar

Heterodyning action

A)

B)

fmax

fmax

200 201 202

Figure 4

Figure 4A shows a channel interferometer transmission spectrum. Figure 4B shows the effective transmission spectrum of amultiheterodyning interferometer formed from many similar channels contiguously filling the input bandwidth.

Trans-mission

Frequency00

fmaxfLO1 fLO2 fLO3

Wide bandwidth spectrum created fromindividually different channel filters

Trans-mission

00

Trans-mission

00

Trans-mission

00

Trans-mission

00

C)

ch

ch

ch

ch

fLO4

Figure 4C shows a non-periodic wideband spectrum formed from many individualized channel spectra.

mathematically in any arbitrary way, and then synthe-sizing the output signal provides powerful flexibility increating a wide variety of signal processes, including theaforementioned non-linear processes which require mix-ing of widely separated frequency components.

The advantage of the invention is that it allows theuse of inexpensive, low bandwidth and low frequency sig-nal processing components such as digital electronics, to

manipulate signals having much higher bandwidth andfrequency. This is especially important for the creationof very long coherent time delays for high bandwidth mi-crowave and optical signals. Such long time delays areneeded to construct velocimeters and range-finders usingshort coherence length illumination, which offer advan-tages of simultaneous unambiguous velocity and rangedetermination, and lack of confusing speckle.

4-6

A great practical advantage of the invention occurs forapplications involving comparison of two signals when aset of reference frequencies can be shared between twomultiheterodyning apparatuses. This comparison can bea correlation between two received signals to determineangle of a radar or optical target, or the correlation of atransmitted and reflected signal to determine range, orthe autocorrelation of a reflected signal previously im-printed with an echo by an interferometer to determinetarget velocity. When two multiheterodyning appara-tuses share the same reference frequencies the compar-ison process is insensitive to the detail values of thosefrequencies. This allows the reference frequencies to driftslowly with time, such as due to thermal drift. This candramatically reduce cost of the apparati by reducing needfor stabilization. Secondly, the channel bandshapes canhave slightly irregular shapes. This reduces the cost ofthe spectrometer.


Figure 1 shows an embodiment of the invention appro-priate for signal processes that create a wideband outputfrom a wideband input, such as delay, interferometry, andfiltering with an arbitrary impulse response. Portions ofFig. 1 are also appropriate for other signal processes. Forthe processes of correlation and autocorrelation, the up-mixing step is omitted and an integral signal summedover all channels forms a net integral output. For theprocess of recording, the real-time up-mixing step is alsoomitted. Instead, it is performed essentially mathemati-cally at a later time. For waveform synthesis, the inputspectrometer 5 and down-mixing steps are omitted.

The multiheterodyning process can be described by fol-lowing a signal through Fig. 1. The wideband input sig-nal 1 having bandwidth βnet is subdivided by the spec-trometer 5 into many separate signals 2, 4, 6, 8 and soon, called band signals. These have smaller bandwidth ofabout βch and different average frequencies. Each bandsignal is processed by a channel in a overall parallel man-ner. Only two of many possible channels are drawn, withthe other channels being analogous. Let fx represent thefrequencies of a generic band signal, and f1, f2 etc. thoseof specific channels. The term “bandshape” refers to thepower transmission versus frequency curve of signals sentinto a particular channel by the spectrometer. It is givenby the output signal power spectrum divided by the inputsignal spectrum so that the transfer function of all thecomponents along the signal path are included, includingthose of the input spectrometer, down-mixer, amplifiersin the channel signal processor, and up-mixer and outputspectrometer, if used. Thus βch is not necessarily set bythe spectrometer, but could be set by other componentsfurther along the channel, whatever is more limiting.

A. Spectrometer

The spectrometer 5 could comprise a diffraction grat-ing or prism, which disperses the signal versus frequencyinto different physical output paths, or a parallel arrange-ment of bandpass filters. It is optimal for the tightest cor-relations and for the most accurate preservation of the in-put signal character that the channel bandshapes fill theinput bandwidth contiguously (shoulder-to-shoulder), sothat there are Nch channels, Nch=βnet/βch. However,sparsely arranged channel bandshapes can be toleratedin many applications, such as correlations. The correla-tion so obtained is less precise than one obtained withcontiguous bands, but it may still be sufficiently preciseto be useful, and is cheaper to achieve. Sparsely filledbandshapes will be the usual case for optical input sig-nals, since the optical bandwidth (1014 Hz) is 5 orders ofmagnitude larger than typical electronics bandwidth (109

Hz), and Nch will usually be quite less than 100,000.The path of a signal through a generic channel will be

described, while sometimes referring to items in the Fig-ures specific to the first channel, for concreteness. Theband signal 2 from the spectrometer is heterodyned toa lower frequency by the down-mixer 7 to form an in-termediate frequency (IF) signal 10, and a quadratureversion 11, called *IF. The heterodyning is done by mix-ing the band signal 2 with a frequency fLO1, which isoptimally located at the edge of the band. (Our analysisassumes fLO < fx. If fLO > fx, the polarities of therequired channel phase shifts to achieve a given effectwill be flipped.) Thus the IF signal could have frequencycomponents ranging from 0 to ∼ βch. Let fIF repre-sent the frequency of a generic component of IF. Let fLO

represent the generic reference frequency and fLO1, fLO2

etc. represent one specific to a particular channel.In Fig. 1, fLO1 is shown generated by an oscillator 12

(typically called a “local oscillator”). However, insteadof having separate signals fLO1, fLO2 etc., the referencefrequencies could be components of a single reference sig-nal, such as provided by a mode-locked oscillator havingmany harmonics. In this case the reference signal couldbe added to the input signal prior to the spectrometer,as shown in the optical embodiments of Fig. 5 and Fig. 6,rather than after the spectrometer. Since fLO and fx areclose together in frequency, the spectrometer could routeboth of these to the same channel where heterodyning issubsequently performed.

B. Down-mixer

Heterodyning is well established in the art. Down-heterodyning can occur if two signals both illuminate anintensity (power) detector or are presented together toa non-linear device (Fig. 7A). Suppose the input signal(band signal) 751 is represented by Exeiωxt and the refer-ence signal component 753 by ELOeiωLOt, then the power(P ) at a detector 754 is |Exeiωx +ELOeiωLOt|2, which be-

4-7

Telescope

Mode-locked laser

Slave

Mode-locked laser

Slave

Referencesignal

Spectrometer

Intensitydetectors

Starlight

+

chnl 1A chnl 1B

Channel1correlation

Net correlation

synchronizing signal

Channel xchannelcorrelator

IF and IF*electricalchannels

Inputsignal

"A"

/4 retarder

45° pol

0° and 90° polarizers

Polarizer

Delaycompensation

Circularly pol. light

Sync generator

∆ n

714

710704

700

702

708

706

712

Inputsignal

"B"

714

703

Figure 5

Figure 5 shows an optical embodiment of a multiheterodyning correlator.

1

2

3

4

5

6

etc.

v-Polarizer h-Polarizer

CCD elementsbehindpolarizers

Chnl#

direction

*IF signals IF signals

Signal

Referencev

h

741740

Figure 6

Figure 6 shows the orientation of polarizers before the inten-sity detectors of the spectrometer in the optical correlator ofFig. 5.

comes

PIF = |Ex|2 + |ELO|2 + 2Re[ExE∗LOei(ωx−ωLO)t]

= constant + ∼ cos (ωx − ωLO)t. (1)

The last term of Eq. 1 is the difference frequency termwhich forms the IF output 750 and has cosine-like charac-ter. The first two terms of Eq. 1 are DC (zero frequency)terms and they need to be excluded from the IF output,otherwise they will masquerade as a legitimate DC termproduced from band signal components having frequen-

Power detection

+-90°

*IF

+IF

fLO

fx

A) Down-Mixer, DC-blocking

DC blocking

Retarder

750

752

759

757From

spectrometer 754

To channel signalprocessor

757

758

753

751

Figure 7

Figure 7A shows a down-mixer creating a down-heterodynedsignal using intensity detection and using a DC-blocking ca-pacitor.

cies near fLO. In Fig. 7A, the DC terms are blocked bycapacitors 757 from the mixer outputs.

In Fig. 7A a quadrature output 752 is formed by het-erodyning the input signal 751 against the reference sig-nal 753 retarded at 758 by -90 (advanced by 90). Thenthe power at detector 759 is |Exeiωxt +ELOei90

eiωLOt|2,which becomes

P∗IF = |Ex|2 + |ELO|2 + 2Re[ExE∗LOe−i90

ei(ωx−ωLO)t]= constant + ∼ sin (ωx − ωLO)t. (2)

The last term of Eq. 2 is the difference frequency termwhich forms the *IF output 752 and has sine-like behav-

4-8

+-270°

-+

+-90°

*IF

+-180°

-+

+IF

fLO

fx

B) Down-Mixer, push-pull arrangement

755

756

Figure 7B shows a down-mixer creating a down-heterodynedsignal using intensity detection and using push-pull subtrac-tion to eliminate the DC component.

ior. These two mixer outputs 750, 752 (IF, *IF) form aso-called “vector” intermediate frequency signal which issent to the channel signal processor (3 in Fig. 1).

In general, the quadrature output (*IF) is producedby heterodyning against a reference signal phase shiftedby 90 relative to the input signal. The relative phaseshift can be +90 or -90 depending on whether *IF isintended to lead or lag IF. A lagging *IF is producedby retarding the input signal, or advancing (retarding bya negative amount) the reference signal, or a combina-tion of both so that the difference in retardations is 90.A leading *IF is produced by the opposite arrangement.(In this specification, the term “retardation” genericallycan have either polarity, and whether a positive or nega-tive retardation is called for needs to be taken from thecontext.)

The choice of whether to retard the input signal or ref-erence signal depends on practical details. For example,for a weak input signal it may be undesirable to inserta retarding element into the input signal path because itmay attenuate the signal, whereas the reference signal isusually strong.

Secondly, it is arbitrary whether to have *IF lead orlag IF, provided the operator is aware that this choicewill flip the polarity of vector rotation ∆θ required toapply to the channel signal process to produce the desiredfrequency shift. The retardation in Fig. 7A is chosen sothat *IF lags IF, so that the *IF signal is analogous tothe imaginary part and not the negative imaginary partof a complex number, and IF represents the real part.

Because it is easy to make a polarity mistake in themathematics that describes vector rotations and channelphase shifts, the reader should use the following opera-tional guideline: the correct polarity of phase shift and

IF rotation to increase the effective delay of the net signalprocess is that which causes the spectrum of the channelsignal process, as mapped to the input spectrum by theheterodyning, to appear to shift to lower frequencies.

It is acceptable that *IF lags or leads IF by an an-gle different than 90, provided it is not a small angle,say not less than 45. For this non-orthogonal case, thesinusoidal coefficients used for the vector rotations aredifferent than that stated by Eq. 9 and Eq. 10. Thecorrect rotational transformation can be found throughstraightforward vector mathematics by re-expressing thenon-orthogonal direction vectors in terms of orthogonaldirection vectors.

A drawback of the down-mixer circuit in Fig. 7A is thatthe DC blocking capacitors 757 also block the legitimateDC components. Figure 7B shows a preferred down-mixer embodiment using a “push-pull” scheme that pre-serves legitimate DC components while eliminating theillegitimate DC components. By retarding the referencefrequency by two amounts 180 apart, such as -90 and-270, or 0 and -180, and by subtracting the associatedintensities, the differences 755 and 756 contain only legit-imate DC terms. The push-pull schematic is analogousto the algebraic expression (A + B)2 − (A−B)2 = 4AB.

Down-heterodyning can also be achieved by replacingthe power detectors in Fig. 7A and Fig. 7B by a non-linear device such as a diode that has a quadratic termin its output versus input behavior.

The heterodyning process can confuse (fx − fLO) sig-nals with (fLO − fx) signals. This confusion is called“aliasing” and is usually unwanted. To avoid aliasing,fLO can be chosen to be on the edge of the band ratherthan in the middle of it. Our analysis assumes fLO is onthe lower edge of the band.

C. Vector signals

The expression of the down-heterodyned signals as apair of signals (IF, *IF), one of which is in quadrature,could be called a vector signal, as opposed to a single IFsignal which could be called scalar.

The vector signals are symbolized in the Figures bya pair of lines, one of which is dashed, such as the pair10, 11 in Fig. 1. The vector expression is useful for ex-pressing phase information for near-DC components ofIF. For example, it is not possible to communicate theboth the phase and amplitude of a DC component of IFby only one signal path. Secondly, even if DC compo-nents are not used, the vector expression of the IF signalallows simple methods for phase shifting all IF frequencycomponents by the same amount, independent of inter-mediate frequency. This is desired to translate uniformlythe spectral behavior contributed by each channel rela-tive to the overall spectral behavior. This is useful forsimulating a dilating net spectrum and hence a dilatingtemporal response for the net device.

This later discussion of phase shifting will center

4-9

Freq.net

ch

fLO1 fLO2 fLO3 fLO4 fLO5

Channels have slope ch

∆ walkoff

∆ dil

= fLines of constant delay

A)

Slope of global line is net

(cycles)

604

∆ lin4

602 net = ch

+∆

601

∆ 4

(1/2) ch ch

0

600 net = ch

"Nominal delay"

0

Figure 8

Figure 8A shows a plot of phase versus frequency of a signal from a signal process having characteristic delay τ .

B)

∆ walkoff

2 ∆ slop

Channel slope ch

Best fitglobal line

fave

608

Figure 8B is a close-up of Figure 8A where the segment rep-resenting a phase-shifted channel meets a global line repre-senting behavior of the net multiheterodyning device.

around Fig. 8A, and a IF phase shifting process thatis independent of IF frequency has the effect of translat-ing the bold line segments 601 called “channel segments”vertically without changing their slope. In point of fact,small changes in the slope can be tolerated, and depend-ing on their polarity they can even help reduce dephasingdue to “walkoff error”. We can call the phase shift non-uniformity over the channel bandwidth an error ∆φslop,and generally we desire it to be less than 0.25 cycle in

order to avoid dephasing effects.To achieve a phase-linear process, it is optimal that

the channel segments match the best fit global line asmuch as possible, such as within 0.25 cycle. In someapplications phase-linearity is not required. For example,to compare two signals being sent through two multi-heterodyning apparatuses, it is the difference in phaseerror between channels “A” and “B” being compared andnot the absolute phase error which is relevant.

D. Channel signal processors

For each channel, the vector signal (10, 11 in Fig. 1)from the down-mixer is applied to a channel signal pro-cessor, such as 3. For processes which will produce ahigh frequency output signal for the net device, such asinterferometry, filtering, time delay or waveform synthe-sis, the channel signal processor outputs a vector signal14 which is sent to the up-mixer 22 to be raised in fre-quency to form a channel high frequency signal 16. Theseare then summed over all channels by 18 to form thenet wideband output 20. For processes such as corre-lation and autocorrelation, the signal processor outputis a slowly varying scalar (also called a time-averagedor integral signal). These are summed over all chan-nels omitting the up-heterodyning step to form the netcorrelation or net autocorrelation. For the process ofrecording, the (IF, *IF) vector channel input is storedfor subsequent analysis at less than real-time and thereal-time up-heterodyning step is omitted. (More pre-

4-10

cisely, the up-heterodyning step is performed essentiallymathematically during the Fourier reconstruction of theinput signal.)

Each channel signal processor has a phase shift param-eter (∆φ) input chosen by the user. This causes the spec-tral behavior for that channel to shift by ∆f = −∆φτch,where τch is the delay characterizing the channel signalprocess. Let ∆φn be the ∆φ for the nth channel. Thechoice of the set ∆φn allows the temporal behavior of theoverall multiheterodyning device to be adjusted precisely.

This phase shifting can be implemented by insertinga retarder into the IF path that delays the IF signalby a phase shift ∆θ ∝ ∆φ which is ideally independentof fIF . The allowed range of fIF may be restricted tothose frequencies (such as by excluding DC components)where the error in ∆θ is tolerable, such as less than 0.25cycle. Alternatively, the phase shifting may be imple-mented by expressing the intermediate frequency signalas a (IF, *IF) vector and rotating that vector throughsinusoidal linear combination by a rotation angle ∆θ,where ∆θ ∝ ∆φ. This has the advantage of workingfor all frequencies, including DC components. Portionsof the channel signal processor responsible for rotationsinclude “vector rotators” and “rotating delays”. An-other method called “post-average weighting” uses unro-tated IF, *IF signals as inputs and applies the sinusoidalweighting to signals formed at the end of a process, suchas after instead of before the time-averaging that occursin a channel correlation process. Another phase shift-ing method is to adjustably retard the reference or inputsignal arriving at the down-mixer.

In single delay processes ∆θ = ∆φ. In multiple delay(multiple echo) processes ∆θ = ∆φ(τm/τch), where τm

is the delay associated with the mth echo. If the *IFleads instead of lags IF, then the polarities of ∆θ areflipped from those stated above. If fLO > fx instead offLO < fx, then the polarities are flipped again.

In cases where channel outputs are summed over chan-nels, the channel outputs are adjusted in gain prior tosumming, to normalize them. This is indicated in theFigures by amplifiers such as 17 in Fig. 1. The normal-ization takes into account the loss of signal through theentire signal path from spectrometer, down-mixer, chan-nel signal processor, and up-mixer and final combiner, ifapplicable. Suppose the signal process was a perfectlyneutral process where the (IF, *IF) vector channel inputdirectly formed the vector channel output. Then opti-mally the gains of the channels would be adjusted sothat the output of the multiheterodyning device wouldbe a coherent replica of the input, as much as possible.

E. Up-mixing

Figure 9A shows an up-mixer, which is responsible forshifting the frequency of the channel output signal up-ward from fIF to, or near to, its original value prior todown-heterodyning. It does this by heterodyning against

x Bandpass

90°

+

x

IF

fLO

fIF+fLO

*IF

A) Up-mixer

From channel outputs High frequency output

Figure 9

Figure 9A shows an up-mixer creating an up-heterodyned out-put using amplitude modulation.

Bandpass

90°

+

IF

fLO

fIF+fLO

*IF

B)

+

+ ( )2

( )2

Up-mixer

Non-linear element

From channel outputs

High frequency output

250

252

Figure 9B shows an up-mixer creating an up-heterodyned out-put using passage of signal and reference through a non-lineardevice.

fLO (or other reference frequency) and bandpass filteringto allow the sum frequency component fIF +fLO to pass.(If fLO > fx, then the difference frequency fIF − fLO isused.)

The quadrature intermediate frequency signal is up-heterodyned against the reference signal retarded or ad-vanced by 90, depending on whether *IF lags or leadsIF. This way the original phase of the input signal canbe restored in the net output.

Usually it is desired to up-heterodyne with thesame reference frequency as was used in the down-heterodyning process, so that spectral components of theinput signal are restored to their original values in thenet output signal, and phase coherence is easy to main-tain. Also, this obviates the problem of stabilizing thefrequency of a second reference frequency relative to thefirst. Practically, it is simplest to use the same referencefrequency for up and down mixing. If it is desired to cre-ate a wholesale shift of frequencies between net input andnet output, such as to compensate for Doppler effects inmoving targets, then this is more practically done by dy-namically changing channel phases ∆φ at a desired beatfrequency.

If it is important to have a phase coherent wide band-width output, then the difference in phase of the reference

4-11

signal at the up-mixer compared to at the down-mixershould optimally be zero or uniform across all channels.For some applications, such as imprinting a comb-filterpower spectrum on a noise source, channel to channelphase coherence is not needed.

Heterodyning is fundamentally a multiplication of theLO signal by the IF signal, and could be accomplished byusing IF to modulate the gain of an amplifier passing LO.Alternatively, the multiplication can be created in effectby passing the sum signal (IF+LO) through a non-lineardevice 250 which has a quadratic term in its behavior, asshown in Fig. 9B.

The bandpass filter 252 blocks the carrier (fLO) anddifference frequency signals (fLO − fIF ). The differencefrequency signals are an aliased signal having a spectralshape which is reversed compared to the un-aliased signal(fIF + fLO). That is, its spectrum is reflected about anaxis at fLO. The aliased signal is usually undesired be-cause any frequency shifts created by the channel phaseshifting will shift the aliased signals in the opposite di-rection. These may form a “ghost” signal which has adifferent spectral character than intended.

In some cases of a pair of multiheterodyning units “A”& “B”, the aliased up-heterodyned signal componentscan be allowed to remain in the net output signal of unit“A”, thereby reducing the cost of the apparatus by omit-ting a filtering step. These cases include when the detect-ing unit “B” has more sparsely arranged bandshapes thatdo not sense the aliased versions of the signals outputtedby illuminating “A” unit.

F. Summation over channels

The last step is to sum the contributions from all thechannels to form the net output. Either the high fre-quency channel signals from the up-mixer are summedto form a net wideband output, or in the case of corre-lations, the channel correlation outputs are summed toform a net correlation.

Figure 10 shows how a spectrometer can be used ina reverse orientation from the conventional usage tocombine many high frequency channel outputs 50 intoone summed output beam 52 having the sum frequency(fIL + fLO) while simultaneously excluding the carrier53 and difference frequencies 54. A slit 56 excludes theunwanted frequencies. If the spectrometer uses a prismmade of a normally dispersive material, then the sumfrequencies will be refracted by a greater angle than theother components, and so the slit should be on the side asshown in Fig. 10. If instead a diffraction grating is used,high frequencies are diffracted less and the slit should beon the opposite side of the carrier beam 53 as shown inFig. 10.

G. Phase Linearity

To discuss the need for channel phase shifting we mustintroduce the concepts of phase-linearity, and tempo-ral/spectral dilation. These concepts are most easy tovisualize for an interferometer. However, they apply tothe other signal processes as well.

Figure 3A shows the ideal power transmission spec-trum of an interferometer T = (1/2)[1+cos (ωτ + const)]where “const” is a constant. The value inside the brack-ets is the phase φ, so that T = (1/2)[1 + cos (φ)], andφ = ωτ +const, where phase is in units of radians. Note,ω = 2πf , so when phase is in units of cycles we have

φ = fτ + const/2π. (3)

The factors of 2π are often omitted from the languagefor brevity, and the symbols ω and f used interchange-ably. The appropriate units for phase is either radians orcycles, depending on the symbol used.

The ideal interferometer is said to be “phase-linear”because φ is a linear function of frequency. If insteadφ = ωτ + const(ω), where const is a non-linear functionof ω, then the interferometer (or other process) is said tobe phase-nonlinear. This concept applies to any processthat can be mathematically expressed in Fourier termseiωt.

A multiheterodyning process is not phase-linear forrandom values of fLO. However, it can be made phase-linear by setting the channel phases ∆φ to a set of values∆φlin approximately given by

∆φlin = fLO〈τch〉, (4)

where 〈τch〉 is the average of τch over all channels, and τch

is the delay characterizing each channel. (The subscript“ch” helps distinguish this value from the delay charac-terizing the net device.) These ∆φlin phase shifts raisethe channel segments 601 of Fig. 8A so they fall along aglobal line 600 having slope 〈τch〉.

It is implied that only the fractional part of a phaseshift expressed in units of cycles need be used, since phaseis periodic. That is, if ∆φ=234.563 cycles, we could use∆φ=0.563 cycles = 202.68.

H. Temporal and Spectral Dilation

When the phase φ is plotted versus frequency, as inFig. 8A, the slope of a line is the delay τ characterizingthe device or process. Slight changes in τ are called a“temporal dilation” and cause a concomitant spectral di-lation of opposite polarity. Because of Eq. 3, f ∼ φ/τ ,so increases in τ at constant φ cause a decrease in f ,and increases in τ holding f constant increases φ. Thuspositive ∆τ is associated with positive ∆φ and negative∆f .

Lines extending from zero to βnet in frequency arecalled “global lines”. The global line slope can obvi-ously be changed by directly altering τ . Alternatively,

4-12

carrier sum

difference

xChannel output6

Prism orgrating

x

x

x

x

x

slit

Channel output5

Channel output4

Channel output3

Channel output2

Channel output1

fLO1fLO2

fLO3 fLO5fLO4 fLO6

Up-mixer using a spectrometer

50

5654

53 52

Figure 10

Figure 10 shows an up-mixer constructed from a spectrometer which combines heterodyned signals from many channels andpasses only the sum frequency components.

and what is interesting about this invention, is that theaverage slope can also be changed by dividing it into seg-ments which are moved as blocks, while preserving theiroriginal, local slopes, which are given by τch. This is whatoccurs in a multiheterodyning processor through channelphase adjustment, where each block represents the regionof input frequencies handled by a channel. Moving thesegments as blocks upward is equivalent to shifting thespectra within each channel lower (leftward) in frequency.(This can be remembered by noting that for lines slopingfrom bottom left to upper right in Fig. 8A, a line lyingabove another line also lies to the left.)

This is illustrated by Fig. 3B through Fig. 3E. Theseshow closeups of different segments of the interferome-ter comb-filter spectrum of Fig. 3A when the interfer-ometer delay is dilated from τ [solid curve] to (τ + ∆τ)[dashed curve]. The spectrum contracts, which locallyshifts segments to lower frequencies by different amounts∆f = −f(∆τ/τ), proportional to segment average fre-quency. The frequency shift can also be expressed as aphase shift called ∆φdil. According to Eq. 3,

∆φdil,n = fave,n∆τ, (5)

in units of cycles, where fave,n is the bandshape’s averagefrequency fave for the nth channel.

Thus the net temporal behavior of the invention can beprecisely altered by an amount ∆τ = ∆φdil/f by phaseadjustments coordinated over many segments (channels).When the device is phase-linear the global line can beconsidered a best fit line through the channel segmentsweighted by the detailed shape of the bands. Becausethis averages the slope over many channels, ∆τ can bemore precisely determined than for a single channel usedalone.

Small delays are achieved by small phase shifts. Forexample, a 10 shift at a 30 GHz channel correspondsto 0.9 ps. Such small delays are needed to resolve shiftsof the autocorrelation peak in a velocimeter for small

velocities. For example, a 2.5 m/s target for a τ=2 msdelay velocimeter produces a round-trip Doppler shift of(2)(2.5 m/s)(2 ms)/(3×108 m/s) = 33 ps.

Although it is easiest to visualize the channel phaseshift induced dilation effect on a phase-linear device, thedilation effect also works on a nonphase-linear device. Italso applies not only to interferometry, but all channelsignal processes, because it is rooted in the expressionof signals as Fourier components eiωt, where ω refers toinput frequencies and does not change significantly acrossa band (because βch βnet).

I. Purpose of Channel Phase Shifting

Thus, two purposes for slightly translating the spec-trum of each channel are: 1) to simulate the dilationof the overall signal process spectrum so that the effec-tive characteristic time τ of the net device is altered ina very precise manner; and 2) to make the net devicephase-linear, if desired. The phase shift applied to eachchannel is a sum of two contributions

∆φ = ∆φlin + ∆φdil (6)

where a term ∆φlin brings the net device into phase-linearity, and a term ∆φdil dilates the effective delay ofthe net device by ∆τ . Approximately,

∆φdil = fave∆τ (7)∆φlin = fLO〈τch〉. (8)

The effective delay can be dilated regardless of whetherthe device is phase-linear or not. This is a key point thatallows the use of reference frequencies that, on a detaillevel, have no special relationship to each other and canbe arbitrary. (On a coarse level, it is optimal to evenlydistribute fLOn across the input bandwidth βnet.)

4-13

It is interesting to point out that for many useful appli-cations a phase-linear device is not required, and in thiscase ∆φlin = 0. A phase-nonlinear device is acceptablefor circumstances where two signals are being comparedthrough multiheterodyning devices sharing the same ref-erence frequencies fLOn. On the other hand, a phase-linear device is desired whenever a signal is to be subse-quently treated by an external apparatus which expectsphase-linearity or cannot share the reference frequencies,e.g., to make a stand-alone delay line which is to outputa coherent replica of the input signal.

Applications needing phase-linearity will also proba-bly need channel bandshapes contiguously filling the in-put bandwidth. Contiguous filling is also optimal forproducing the least ambiguous and narrowest correlationand autocorrelation peaks. However, sparsely filling theinput bandwidth is more economical, and can be toler-ated by multiheterodyning apparatuses sharing referencefrequencies.

J. Rotation of the IF Vector

The channel processor is designed to shift its spectrumby an amount ∆f = −∆φ/τch. That is, positive phaseshifts produce negative frequency shifts. In general, asignal process may use delayed and undelayed compo-nents of the (IF, *IF) signal vector. A spectral shift canbe created by using, for the delayed components, a ro-tated version of the (IF, *IF) vector rotated by an angle∆θ. This is called (IF′, *IF′) and given by a rotationaltransformation

IF ′ = IF cos ∆θ +∗ IF sin ∆θ (9)∗IF ′ = −IF sin ∆θ +∗ IF cos ∆θ. (10)

Note, this method works even for the DC components of(IF, *IF).

If the (IF, *IF) vector is represented by ∼ eiωt, thenthe rotation specified by Eq. 9 & Eq. 10 is the same asmultiplying that term by a phasor (e−i∆θ), a clockwiserotation in the complex plane, or a retardation. (To testthis, set ∆θ=90 and we see that IF becomes *IF, whichagrees with our assumption that *IF lagged IF.)

Figures 11A through 11F show how rotation of the IFvector works for the case of an interferometer. The sim-ple interferometer shown in Fig. 11A uses IF for boththe undelayed 630 and delayed 632 branches and hasthe power transmission spectrum shown in Fig. 11B,T = (1/2)[1 + cos (ωτ)]. Figure 11C shows an interfer-ometer using for the delayed branch the lagging quadra-ture *IF signal instead of IF. The corresponding spec-trum T = (1/2)(1 − sin ωτ) shown in Fig. 11D is shiftedto lower frequencies by an amount equivalent to 90 inphase. Thus to translated the spectrum by an adjustableamount, the delayed branch should use an IF signal ro-tated by an adjustable amount.

This is accomplished by the circuit Fig. 11E, whichimplements the first equation of the rotational transfor-

mation Eq. 9. The corresponding spectrum is shown inFig. 11F. The mathematical derivation of the spectrum isas follows. Let the undelayed branch be E(t) = eiωt andthe delayed branch E(t−τ) = e−i∆θe−iωτeiωt, where thephasor e−i∆θ represents the adjustable rotation. Thenthe power spectrum is |E(t) + E(t − τ)|2 and henceT = (1/2)[1 + Re(ei∆θeiωτ )] = (1/2)[1 + cos (ωτ + ∆θ)],where “Re” means “real part of”. Hence positive valuesof ∆θ shift the spectrum (dashed) to the left (loweringfrequencies). Thus we want ∆θ = ∆φ.

K. Channel Interferometer with Vector Output

The channel interferometer of Fig. 11E produces ascalar output. This is sufficient for phase insensitive ap-plications, which care only about the power spectrum,such as velocimetry. However, to be mathematically com-plete and for applications where the accurate propagationtime of the undelayed portion of the signal through theinterferometer matters, a vector output should be pro-vided, so that the correct complex description of the in-put signal can be propagated through the channel signalprocessor. For example, in a combination process wherecorrelation follows interferometry, such as to determinerange as well as determine velocity, a vector channel out-put is desired from the channel interferometer.

Figure 12A shows a channel interferometer producinga vector output. This could be called a “vector interfer-ometer”. The circuit inside the bold oval 650 could becalled a “Rotator” and is responsible for rotating the IFvector signal by ∆θ, to produce a rotated vector (IF′,*IF′). Nominally we would use ∆θ = ∆φ. The value ofτ used in the delays 651 is the parameter τch used in thephase shift analysis.

An interferometer is a special case of an arbitrary fil-ter where there is either a constant spacing between echosor a small number of distinct spacings. A spacing valueis called the interferometer delay τ , and it is the mostimportant parameter controlling the interferometer be-havior. For purposes of forming autocorrelations, it isoptimal to have as few distinct τ values as possible, ide-ally a single value. Otherwise there could be a dilutionand confusion of autocorrelation peaks, since there willbe a peak associated with each distinct τ value. Multipleτ values can be tolerated if they are significantly differentthan each other (separated by more than several timesthe coherence time, 1/βnet).

The most common type of interferometer is a two-pathinterferometer analogous to the Michelson interferome-ter, where one echo of equal amplitude to the undelayedsignal is produced. This echo could have a positive ornegative polarity amplitude compared to the undelayedsignal, producing complementary outputs analogous tothe two complementary arms of a Michelson interferom-eter. Another useful kind is a recirculating interferome-ter which generates an infinite number of equally spacedechos having geometrically decreasing amplitudes.

4-14

A) +

Delay

IF

C) +

Delay

IF

*IF

B)

0 Frequency

Trans-mission

0

1/

(1/2)[1+cos( )]

D)

0 Frequency

Trans-mission

0

(1/2)[1-sin( )]

Channel output

630

632

Figure 11

Figure 11A shows an interferometer without phase shift. Figure 11B shows the power transmission spectrum of the interfer-ometer of Fig. 11A. Figure 11C shows a 90 phase-shifted interferometer using quadrature IF signal for the delayed branch.Figure 11D shows the power transmission spectrum of the interferometer of Fig. 11C.

+

Delay

*IF

IF cos(∆ )

sin(∆ )

+

∆E)

F)

0

-∆∆f =

Frequency

Trans-mission

0

Phase adjustments to slightly shift effective channel filter position

Channel output

IF signal rotated by ∆ ∆

Figure 11E shows an interferometer phase-shifted by an adjustable amount. Figure 11F shows the spectrum associated withthe interferometer of Fig. 11E having an adjustable frequency shift.

+

Delay

cos(∆ )

sin(∆ )

-sin(∆ )

cos(∆ )

*IF

IF

Channel output

Delay

+ Channel *output

A)

IF'

*IF'

Rotated by ∆

Channel Interferometer

+

+

Rotator 650

∆ ∆

651

651

Figure 12

Figure 12A shows a channel signal processor which is an interferometer having delay τ , vector rotation ∆θ, and forming avector output.

4-15

Delay

cos(∆ )

sin(∆ )

-sin(∆ )

cos(∆ )

*IF

IF

Channel output

Delay Channel *output

B) Channel Delay

+

+

Rotator

652

652

650

IF'

*IF'

∆ ∆

Figure 12B shows a channel signal processor which is a delay of duration τ and vector rotation ∆θ.

Data storage*IF

IF Data storage

C) Channel Recorder

Data storage

Data storage Channel output

Channel *output

D) Channel Synthesizer

Figure 12C shows a channel signal processor which recordsthe IF and quadrature IF signals. Figure 12D shows a channelsignal processor which synthesizes the channel and quadraturechannel outputs.

An interferometer can be used to form an autocor-relator, when the time-averaged power passing throughthe interferometer is measured versus characteristic de-lay. For this purpose it is optimal that all the echos beequally spaced. A two-path interferometer is the optimalinterferometer to use because it can yield a single-peakautocorrelation. In contrast, multiple echo interferome-ters yield autocorrelations having multiple peaks. Theseare still useful in situations (such as where small velocitychanges are being measured) where the central peak canbe distinguished from the other peaks.

A pair of matched interferometers can be used forvelocimetry using wideband illumination (Fig. 2A andFig. 18). The “A” interferometer imprints a comb-filterspectrum on a wideband source 100, which is to say itimprints echos having characteristic delay τA. This im-printed beam 104 can illuminate a target. A secondinterferometer (“B”) having nearly identical character-istics observes the waves 106 reflected from the target.The time-averaged power passing through the second in-terferometer is measured versus “B” interferometer delayto essentially form an autocorrelation. The peak locationof this autocorrelation shifts with target velocity (and is

proportional to τA).

Strictly speaking, for this velocimetry purpose otherkinds of matched filters could be substituted for the twointerferometers. However, compared to a two-path inter-ferometer these may produce weaker autocorrelations, orthe behavior may be more complicated to analyze math-ematically. The two-path and recirculating interferome-ters are the two most useful kinds of filters for widebandvelocimetry.

L. Channel Delay Line

A channel delay unit that produces a vector outputand has the signal rotation property is shown in Fig. 12B.This could be called a “rotating delay” or a “vector de-lay”. Delay units are fundamental building blocks forconstructing filters. Nominally ∆θ = ∆φ.

A method of implementing the actual delaying of theIF or *IF signal, indicated by the boxes such as 652 usedthroughout the Figures labeled “Delay τ”, include dig-ital and analog methods. A digital delay line consist-ing of an analog-digital converter (A/D), shift register,and digital-analog converter (D/A) can easily create sev-eral millisecond delays. These would have a bandwidthroughly half the clock frequency used to drive the shiftregisters. Analog methods include conversion of IF sig-nal to an acoustic wave which propagates a distance andthen is detected, or use of the IF signal to modulate theintensity, color, phase or polarization state of an opticalsignal sent through a long fiber.

4-16

LO1 LO2 LO3 LON

Multiheterodyning Interferometer

Net autocorrelation (velocity information)

Widebandsource

A)

Shared set of referencefrequencies

∫dt2

, sinusoidal

406

400 401 402 403

408

+

+

"B" unit

"A" unit

∆ n,B

∆ n,A Multiheterodyning Interferometer

weighting and gainnormalization

104

106

100

Figure 18

Figure 18A shows a double interferometer velocimeter using two matched multiheterodyning interferometers sharing referencefrequencies.

Frequency

Transmission

Irregular channel bandshapes, frequencies

00

B)

fLO4

Not necessarily commensurate

412 414418416

fLO1 fLO2 fLO3

Figure 18B shows irregular channel bandshapes, frequency locations and non-commensurate comb-filter peaks.

M. Channel Recirculating Interferometer

Figure 13A shows a circuit which creates a recirculat-ing interferometer, which is a kind of filter analogous toan optical Fabry-Perot interferometer. This has the im-pulse response shown in Fig. 13B, an infinite series ofgeometrically decreasing echos 671, 673 etc. separatedby interval τ . The amplitude of each subsequent echois R times the preceding echo. The Fourier power trans-form of the impulse response Fig. 13B is the transmissionspectrum Fig. 13C. This has the same periodicity 1/τ asthe sinusoidal comb-filter but has narrower peaks. Thecloser R is to unity, the narrower the peaks are relative tothe spacing between peaks. The circuit Fig. 13A recircu-lates the vector IF signal. For each round trip around thecircuit and through the block 672, the vector signal 676

is delayed by τ , rotated in phase by ∆θ, and its ampli-tude multiplied by factor R (which must be <1 to avoidoscillation). Nominally ∆θ = ∆φ.

N. Channel Arbitrary Filter

To create an arbitrary filter one defines its impulse re-sponse, since the impulse response and spectral responseare Fourier transforms of each other. Let the first spike300 of an impulse response (Fig. 14B) define unity am-plitude. Subsequent spikes 301-304 are called echos andhave user-chosen amplitudes and arrival times. The ar-rival times are not restricted to be uniformly spaced andcan have any time value. Then one constructs a circuit(Fig. 14A) using one or more rotational vector delays306, 307 etc., shown in bold ovals, having τm and Rm

4-17

+

Delay

cos(∆ )

sin(∆ )

-sin(∆ )

cos(∆ )

IF'

*IF'

*IF

IF

R

Channel output

Rotatedversions of IFand *IF

Delay R

+

+

+ Channel *output

+

+

672

676

∆ ∆

Figure 13

Figure 13A shows a channel signal processor which is a recirculating interferometer having a set amount of signal rotation,delay and attenuation per circuit.

B) Impulse response

Time

Amplitude

0

1

R

0

∆ 2∆ 3∆ 4∆ etc.0∆

670

671673

Effective

Figure 13B shows the impulse response for a recirculatinginterferometer.

values for the mth echo which will create the set of echosin the output described by the impulse response. Eachrotational delay 306, 307 etc. represents the circuit 672shown in the bold oval of Fig. 13A.

Figure 14A shows an example circuit which would cre-ate the impulse response Fig. 14B having 4 echo spikessubsequent to the first spike. The paths comprising adashed and undashed pair of lines represent vector sig-nals. The manipulations (delay, addition, subtraction)act analogous to vector arithmetic in that they are per-formed on IF and *IF in parallel. Figure 14C shows amore complicated channel filter using interwoven recir-culation.

Transmission SpectrumC)

Frequency0

Trans-mission

1/

-∆∆f =

Figure 13C shows the power transmission spectrum for therecirculating interferometer.

In order to slightly shift a channel filter spectrum infrequency, the phases ∆θm associated with the mth spikeof a filter impulse response Fig. 14B must be changedproportional to the arrival time of the spike in the im-pulse response. That is, ∆θm = ∆φ(τm/τch), where ∆θm

is the phase shift of the mth rotating delay unit havingdelay τm, and τch is the characteristic delay of the chan-nel filter. This can be set to one of the τm values, suchas the echo having the largest amplitude.

4-18

+

∆ 4 , 4, R4

Channel vector input

Channelvector output

Phase proportional todelay time∆ m = m (∆ ch)

A)

∆ 3 , 3, R3

∆ 2 , 2, R2

∆ 1 , 1, R1

306

307

Figure 14A shows a channel signal processor constructed ofvector delays that realize a four-echo impulse response.

B) Impulse response

Time

Amplitude

0

1

R3

0

300301

302

303

304

Figure 14B shows a four-echo impulse response.

O. Channel Autocorrelator

An autocorrelation is a correlation of a signal with it-self. It is useful to define an autocorrelation AC evaluatedat τ of a function f(t) as

AC(τ) = Re∫

dt f(t)f∗(t − τ). (11)

The integral represents time averaging, which occursover a duration much longer than 1/βch. Time averag-ing can be performed by the invention either before orafter up-heterodyning. However, practically it is sim-plest to do it before heterodyning and thereby eliminatethe up-mixing step. This is done by summing the chan-nel outputs produced by each channel autocorrelator toform the net autocorrelation output.

A channel autocorrelator is shown in Fig. 15A. Itsoutput 776 is a sum of two separate autocorrelations,which could be called intermediate outputs or interme-diate autocorrelations, one (777) from the IF signal, and

a quadrature version 778 from *IF. Note, in some appli-cations the quadrature intermediate output 778 can beomitted because signals which show correlations for thereal part of the their Fourier spectrum often show thesame correlation in the imaginary part. For example, invelocimetry applications, the illuminating interferometer(“A” in Fig. 2A or Fig. 18) creates the same comb-filterpower spectrum for the real and imaginary parts of signalsent to the target.

It is convenient to implement the multiplication func-tion 789 by passing a sum or difference of the two signalsto be multiplied through a squaring function, or non-linear device, or power measuring device. Figure 16Cshows a simple (“lopsided”) circuit that does this, analo-gous to the formula (A+B)2 = 2AB +(A2 +B2). Whenthis is done, essentially the autocorrelation comprisesmeasuring the time-averaged power passing through avector interferometer 781, as shown in Fig. 15C, and ig-noring any delay independent constants. That is, thetime-averaged power of a signal f(t) passing through atwo-path interferometer of delay τ is

P1(τ) =∫

dt |f(t) + f(t − τ)]|2 = 2|f(t)|2 + 2AC(τ)

(12)

where |f(t)|2 is the undesired constant, and the secondterm is the desired autocorrelation.

A method which automatically deletes the undesiredconstant term is the push-pull method. This can be ap-plied either before or after time-averaging. When appliedbefore, it comprises a method of multiplication at 789analogous to the formula (A + B)2 − (A − B)2 = 4AB.This form of push-pull is shown in Fig. 16B. When ap-plied after, two separate interferometers analogous toFig. 15C are required, except one of the interferometersubtracts the echo at 785 instead of adding it at 785 tothe undelayed branch. Then in addition to the time-averaged power P1 of the first interferometer, the time-averaged power P2 of the second interferometer is found.This is

P2(τ) =∫

dt |f(t) − f(t − τ)]|2 = 2|f(t)|2 − 2AC(τ).

(13)

Then the two time-average power signals are subtractedand divided by four to form the autocorrelation

AC(τ) = (1/4)[P1(t) − P2(t)]. (14)

This method automatically suppresses the |f(t)|2 term.For clarity, circuits using the lopsided method are shownin the Figures, but it is implied that the push-pullmethod can be substituted wherever the lopsided methodis used.

4-19

Ra

+ ∆ 1 , 1, R1 + ∆ 2 , 2, R2 ∆ 3 , 3, R3 +

Rb

Rc

C)

Figure 14C shows a channel signal processor constructed of many recirculating paths which will produce a more complicatedimpulse response.

Delay

Delay

*IF

IF x

x

May be omitted in manyapplications

Rotator

778

776

789

A) Channel Autocorrelator

780

∆*IF'

IF'

+ Channelautocorrelationoutput

∫ dt

∫ dt

779

777

∆ ∆

Figure 15

Figure 15A shows a channel signal processor which is an autocorrelator having delay τ and signal rotation ∆θ.

B) Illuminating interferometer of a velocimeter

+Delay Noisegenerator To Up-Mixer

Channel 1

+Noisegenerator To Up-Mixer

Channel 2

+Noisegenerator To Up-Mixer

Channel 3

772770

IF 774

Delay

Delay

Figure 15B shows channel waveform synthesis using multi-ple channel noise generators instead of spectrometer-producedsignals.

P. Sinusoidal Weighting After Time-averaging

For processes such as correlations that output a time-averaged signal, there is another way of phase shiftingthe channel spectrum that avoids having to rotate the(IF, *IF) vector. Instead the unrotated IF and *IF are

used as inputs. This could be called the “post-averageweighting” technique. In this technique the sinusoidallinear combinations are applied after and not before thetime-averaging. Therefore they operate on very low fre-quency, which reduces cost of components performing thearithmetic and makes it economical to calculate them fora variety of ∆θ values in parallel. This allows economicalevaluation of the correlation over a range of delays, in asnapshot, on the same set of IF vector data.

The mathematical basis for this is now given, for thegeneral case of two-signal correlation. Let the two func-tions to be correlated be

f(t) = fr + ifi (15)h(t) = hr + ihi (16)

where fr, fi, hr, hi are the real and imaginary parts off(t) and h(t) and are directly analogous to the interme-diate signals and their quadratures IFA, *IFA, IFB and*IFB , shown in Fig. 17 and as 813-816 in Fig. 16A. Thena correlation between f(t) and h(t) is

Cor(τ) = Re∫

dt f(t)h∗(t − τ). (17)

4-20

*IF

IF +

+

C) Channel Autocorrelator


∫ dt

∫ dt

( )2

( )2

Time-averagedpower

Vector interferometer

∆ Delay

781

∆ ∆

785

Figure 15

Figure 15C shows a channel autocorrelator formed by measuring the time-average power passing through a vector interferometer.

D) Channel Autocorrelator


∫ dt cos∆

∫ dt cos∆

+Delay ( )2

+Delay ( )2

InterferometerTime-averaged

power

IF

*IF

782784

783784

Sinusoidalweighting

Figure 15D shows a channel autocorrelator where the phaseshifting is applied after the time averaging.

Suppose we want to evaluate Cor(τ) for small devia-tions around τ=0 (for this demonstration it is immaterialwhat the coarse value of τ is). Then let τ be ∆τ . Thenbecause we can express a complex function in terms ofFourier components, we have

h(t − ∆τ) → e−iω∆τh(t) = e−i∆θh(t) (18)

where we used the assumption that βch βnet so thatthe frequency ω in Eq. 18, which refers to frequencies ofthe input signal, is almost constant across a given chan-

nel. Thus ω∆τ can be approximately represented by asingle value ∆θ, for each channel. Equation 202 becomes

h(t − ∆τ) → e−i∆θh(t) = h′r + ih′

i (19)

where

h′r = hr cos ∆θ + hi sin ∆θ (20)

h′i = −hr sin ∆θ + hi cos ∆θ. (21)

This is the rotational transformation previously discussedto rotate the IF, *IF vector. Thus we have

Cor(∆τ) = Re∫

dt f(t)h′ ∗ (t) (22)

where h′(t) is the rotated form given by Eq. 20 andEq. 21. Equation 206 is what is evaluated by the correla-tion circuit Fig. 16A, and by the autocorrelation circuitFig. 15A (when h(t) = f(t)).

Now we will re-express Eq. 22 in another form thatplaces the sinusoidal linear combinations Eq. 20 andEq. 21 outside the time integrations instead of inside.Substituting Eq. 20 and Eq. 21 into Eq. 22 yields

Cor(∆τ) = Re∫

dt (fr + ifi)(hr cos ∆θ + hi sin ∆θ) − i(−hr sin ∆θ + hi cos ∆θ) (23)

which, after taking the real part of, becomes

Cor(∆τ) = cos ∆θ

∫dt frhr + cos ∆θ

∫dt fihi + sin ∆θ

∫dt frhi − sin ∆θ

∫dt fihr. (24)

Equation 206 shows that the channel correlation can be computed without rotating the IF vector, by first com-

4-21

*IF

IF

Channel 1B

Channel 1A

A) Channel Correlator812

810

Delay

Delay

*IF

IF x

x+

Rotator

∆

820

*IF'

IF'

813

814

815

816

∫ dt

∫ dt

∆ ∆

Figure 16

Figure 16A shows a channel signal processor which is a correlator between A & B channels, having delay τ and signal rotation∆θ.

B)

( )2

Non-linear element

+-+

+

-( )2

Timeaverager

Delay

IF

Correlationoutput

Pushpull method of multiplication

(A+B)2 - (A-B)2 = 4AB

Figure 16B shows that the multiplication function used in Figure 16A can be accomplished by a push-pull arrangement of anon-linear element.

+ ( )2Time

averager

Delay

IFCorrelationoutput

C) Lopsided method

(A+B)2 = 2AB + constant

816818

Figure 16C shows the simple use of a non-linear element to create a correlation signal having an offset.

4-22

D) Channel Correlator

+ Channelcorrelationoutput

Delay

IFA x ∫ dt cos∆

Delay

x ∫ dt cos∆

Delay

x ∫ dt sin∆

Delay

x ∫ dt - sin∆

IFB

*IFA

*IFB

IFA

*IFB

*IFA

IFB

820

821

822

823

∆ ∆

Intermediatecorrelationsformed

Sinusoidalweighting

Figure 16

Figure 16D shows a channel correlator where the phase shifting is applied after the time averaging.

puting the intermediate correlations

Cor1 =∫

dt frhr

Cor2 =∫

dt fihi

Cor3 =∫

dt frhi

Cor4 =∫

dt fihr (25)

that only involve the IF and its quadrature. That is usingvarious combinations of IF against IF, IF against *IF, *IFagainst *IF etc.. Then forming sinusoidal linear combi-nations of these intermediate correlations to produce thechannel correlation.

Now for the case of an autocorrelation, fr = hr andfi = hi, and the last two terms of Eq. 24 cancel, leaving

AC(∆τ) = cos ∆θ

∫dt frfr + cos ∆θ

∫dt fifi (26)

The circuit of Fig. 15D embodies Eq. 26, showing theautocorrelation calculated from a sum of two intermedi-ate autocorrelations 782 and 783, with a sinusoidal co-efficient 784. Note that in many situations, an inputsignal will show the same autocorrelation behavior in its

real part as in its imaginary part. Thus, for economythe

∫dt fifi term, which is calculated from *IF, can be

omitted, and the *IF term does not have to be produced.This simplifies the down-mixer apparatus by eliminatingthe quadrature portion of it, and eliminating the need forretardation 758.

A important advantage of applying the sinusoidal lin-ear combination after the time averaging is that it makesit economical to calculate the autocorrelation or corre-lation for a range of delay values τ , without having toactually scan τ in real-time. Since the sinusoidal arith-metic is being performed on a very low frequency signal (atime-average), the arithmetic can be easily and rapidlyevaluated for many values of ∆θ. This can produce asnapshot evaluation of AC(τ) or Cor(τ) over the rangeof τ accessible by coordinated channel phase shifting, arange τ±∆τmax, where ∆τmax is defined in a later discus-sion and is limited by walkoff errors. This is much morerapid than actually scanning the channel delay, whichwould require dwelling on each delay value long enoughfor the time-averaging to occur.

Q. Channel Correlator

Figure 17 shows a multiheterodyning correlator whichcorrelates two wideband input signals #A and #B. Ap-

4-23

Spe

ctro

met

er

Widebandinput signal#A

DownMixer Channel

correlatorLO1

f1A

fLO1

IFA

Similarly for allchannels

Correlating atintermediate frequency

Heterodyning todown-shift frequency

+

Mul t ihe te rodyn ing Cor re la tor

Spe

ctro

met

er

Widebandinput signal#B

DownMixer

f1B

Netcorrelation

corr1

corr2

corr3

f2A

f3A

f2B

f3B

∆ 1

*IFA

IFB

*IFB

∆ 2

792

790

794

791

793

798

796

Channelcorrelator

Phasecompensator

Gain normalization

797

795

799

Figure 17

Figure 17 is a schematic of a multiheterodyning correlator.

Multiheterodyner

Shared set of localoscillators

A)

Channel-by-channelcorrelator + Net correlation

yields targetrange

LO1 LO2 LO3 LON

Phase compensators

504500

502506

508

510

"A" unit

Multiheterodyner

"B" unit

Widebandsource

∆ n,B

∆ n,A

Gain norm.

Figure 19

Figure 19A shows a correlator using two matched multiheterodyning units sharing reference frequencies.

Set of oscillators or mode-locked source

LO1 LO2 LO3 LON

Set of oscillators or mode-locked source

LO1 LO2 LO3 LON

Synchronizing signal

Long distance

B)

530

534 532

To first unit To second unit

Synchgenerator

Figure 19B shows synchronization of reference signals for two widely separate multiheterodyning units.

4-24

plications include correlating natural wideband signalssuch as starlight (Fig. 5), radio emitting stars, and artifi-cially created signals such as illumination created a rangefinding radar, sonar or lidar (Fig. 2B and Fig. 19), or il-lumination from the “A” interferometer of a velocimeter(Fig. 2A and Fig. 18). Each channel pair 1A and 1B,2A and 2B etc., are correlated separately by the channelcorrelator units 791, 793 etc. to produce channel corre-lation outputs 790, 792 etc. These outputs are summedto form the net correlation output 794. The same refer-ence frequency fLO1 is shared between channel 1A and1B, and similarly for the other reference frequencies. Thebandshapes produced for the A and B versions of eachchannel should optimally be as similar as possible. Itmay be possible in some embodiments to use the samephysical spectrometer to implement spectrometers 796,798 by having the A inputs and outputs displaced fromthe B input and outputs, displaced perpendicular to thewavelength direction.

The phase compensator 795 allows adjustment of thedifference in arrival time of the reference signal betweenthe down-mixers 797 and 799. This could correct fordifferences in propagation times in the cables routingthe reference signal. These also provide an alternativemethod implementing the channel phase shifts, by pro-viding another method of rotating the intermediate fre-quency signals, either rotating IF alone or the vector (IF,*IF). This method can be applied for all the signal pro-cesses, not limited to correlations. Equation 1 shows thata phase shift imparted to the reference or input signalswill manifest that same phase shift for the correspond-ing down-heterodyned (IF) signal, and similarly for Eq. 2and *IF. (The retardations could be applied to either in-put signal or reference signal, but for concreteness let usassume they are applied to the reference signals). Henceretarding the arrival time of the reference signal at “B”down-mixer 799 relative to “A” down-mixer 797 causes aphase shift equivalent to changing |∆θ| and hence |∆φ|.In an interferometer process, for example, instead of ob-taining the rotated vector (IF′, *IF′) from a rotator block(650 in Fig. 12A), the rotated (IF′, *IF′) signals could beobtained from a down-mixer unit having an adjustablyretarded reference signal. In order to rotate by a posi-tive ∆θ, the retardation of the reference signal would benegative ∆θ. In optical embodiments of the multihetero-dyning processor, the adjustable retardance of the refer-ence beam or input beam can be implemented electro-optically. Collectively (for all channels) changing the ar-rival time of the reference signal to the down-mixer, suchas by changing a path length, is a method of changingthe net delay of the correlating device.

Figure 16A shows a channel correlator. It comprisestwo independent correlators whose outputs are summed,one from the IF signal and one from the *IF signal. Themultiplication function 812 could be implemented by acoincidence gate, such as when photon-counting detectorsare used. Nominally ∆θ = ∆φ.

Correlating more than two inputs (such as A, B, C,

D, E) can be done by constructing a circuit analogous toFig. 16A that has multiple vector inputs and implementswhatever mathematical definition of a multi-variable cor-relation the user desires. This includes finding the timeaverage of the square of the sum of the inputs.

Figure 16D shows how the correlation can be per-formed using the post-averaging weighting method, byforming a sinusoidal linear combination of intermediatecorrelations 820, 821, 822, 823, as described by Eq. 24.Each intermediate correlation is a correlation betweencombinations of IF and *IF. This method is advanta-geous when the down-heterodyned signal is detected by aphoton-counting system, and a coincidence gate used forthe multiplication function of the correlator. This way,the sinusoidal linear combination can be applied to cu-mulative output of the coincidence gate, which is a slowlychanging signal, rather than the higher bandwidth pulsesconstituting the IF signals.

R. Channel Recorder

Figure 12C shows a channel recorder which stores theIF vector signals for later use. At the time of measure-ment, the frequencies fLOn, relative phases, and ampli-tudes of the reference signal are also measured. The inputsignal can be recreated from this stored information byFourier reconstruction.

S. Channel Waveform Synthesizer

For the process of waveform synthesis, vector channeloutputs sent to the up-mixer are ersatz and can be takenfrom stored data, or from some real-time channel processnot based on a high frequency (∼ βnet) input. For exam-ple, Fig. 15B shows how to generate a high bandwidthnoise signal having a comb-filter power spectrum , such asfor use in velocimetry (Fig. 2A). Each channel 774 couldhave a noise generator 770 that passes through a chan-nel interferometer 772. That is, instead of using a singlehigh bandwidth noise generator as a high frequency input(1 in Fig. 1) passing through a spectrometer 5, the noisecould be generated at the channel level in a parallel man-ner by many lower bandwidth noise generators. Thus aspectrometer is not needed. Furthermore, this randomsignal 770 does not have to be vectorized for velocimetryapplications, since the phase of the signal 104 leaving thefirst interferometer 102 is immaterial.

T. Combination Signal Processes

A combination of channel signal processes can be ap-plied to the channelized intermediate frequency signalsin an overall parallel manner to implement a combinedwideband signal process. The combinations can include

4-25

the channel signal processes already discussed, and arith-metic operations such as adding and subtraction of in-put signals. Let s1, s2, s3, etc. be channel signal pro-cesses, called component processes, and f(s1, s2, s3, . . . )be a network of these, where outputs of any componentprocess sx could be used as part of the input of any othercomponent process sy. That is, we are not restricted tosimple sequences s1 to s2 to s3 etc. Then this will forma multiheterodyning device that acts on the widebandinput signal analogous to f(S1, S2, S3 . . . ), where S1, S2,S3 etc. are the individual effect on a wideband signal bys1, s2, s3.

In other words, going from the output of one multi-heterodyning apparatus to the input of another we canomit the steps of up-heterodyning, summation over chan-nels, then spectral decomposition into the same chan-nels, and down-heterodyning. This is because there is nopoint to up-heterodyning and then immediately down-heterodyning, and no point to summing over channelsand then immediately decomposing into the same chan-nels. Effectively we can move these steps to outside thenetwork f(s1, s2, s3, . . . ). Generally, it is advantageousto keep the wideband signal information expressed as aparallel set of intermediate frequency signals as long aspossible, postponing the steps of up-heterodyning andsummation over channels.

U. Calculating Desired Channel Phase Shifts

1. Plotting phase versus frequency

The most important parameter describing the behav-ior of a signal process is the phase φ = fτ . Thus it isextremely useful to refer to Fig. 8A, which plots φ versusf . A phase-linear device having delay τ is representedby a line φ = fτ . The long thin lines 600 and 602 are“global” lines φ = fτnet representing the effective net de-lay τnet of a multiheterodyning device. (The difference inslopes between lines 600 and 602 has been greatly exag-gerated relative to their absolute slope for clarity.) Thedelay created by an individual channel is a bold short line(such as 604) of horizontal extent βch, having slope τch,and called a “channel segment”. These τch values canhave individually different values (τch,n) from channel tochannel. The global lines are best-fit lines through allthe channel segments, weighted by the detailed shape ofthe channel bands.

Since phase is periodic with periodicity of 1 cycle or2π radians, the channel segment and global lines can beequivalently translated vertically any integer number ofcycles on Fig. 8A.

2. An Initially Phase-nonlinear Device

When no channel phase shifts are applied, the channelsegments initially lay on the horizontal axis, with their

left (low frequency) ends at coordinates (f=fLOn, φ=0),and having slopes τch,n. However, due to the periodicityof phase, the channel segments can be raised vertically byadding any integer number of cycles. Then a global linewhich is a best-fit line through all the channel segmentsis attempted to be found. However, for arbitrary valuesof fLOn, it is unlikely that there will be a very good fit,as there will be random phases, some too high and sometoo low. Thus, a non-phase linear device will probablyresult from arbitrary choice of fLOn.

3. Bringing it into Phase-linearity

Bringing the net device into phase-linearity is equiva-lent to raising each channel segment by an amount ∆φlin

so that the “center of mass” of the channel segment atfave falls along a global line. The global line 600 has a“nominal” delay τnet = 〈τch〉. Global line 602 describesa slightly larger delay, τnet = 〈τch〉 + ∆τ , created by al-tering the channel phase shifts by an additional amount∆φdil.

V. Creating Small Changes in Delay

The ∆φdil phase shifts can be used to create smallchanges ∆τ in τnet while holding τch constant for all chan-nels. Phase shifts cannot be used to create large changesin τnet, because then the slope of the channel segmentsbecomes too different from τnet, creating walkoff errors∆φwalkoff (608 in Fig. 8B). The walkoff error is approx-imately

∆φwalkoff = (1/2)βch(τnet − τch,n)≈ (1/2)βch∆τ. (27)

When the walkoff errors grow beyond approximately±1/4 cycle, dephasing of the channels makes the multi-heterodyned output incoherent, and thus no-longer use-ful for double interferometer velocimetry or correlationmeasurements. Thus the maximum practical dilation is

∆τmax ≈ (1/2)/βch. (28)

To create a larger delay change than ∆τmax requireschanging τch, in addition to re-adjusting the channelphasing.

Note that this result assumes ∆θ and hence ∆φ is inde-pendent of intermediate frequency. A positive frequencydependence to ∆θ and ∆φ would increase the slope ofthe segment and change the walkoff error. This wouldreduce walkoff and raise ∆τmax for positive ∆τ , and in-crease walkoff and reduce ∆τmax for negative ∆τ .

IV. MASTER PHASE EQUATION

To achieve a coherent delay, we find the phase shifts∆φn that put the “center” (located at fave) of each chan-

4-26

nel segment along a global line having a slope τnet. Thus

(1/2)βchτch,n + ∆φn = [fLOn + (1/2)βch]τnet (29)

which becomes

∆φn = [fLOn + (1/2)βch]τnet − (1/2)βchτch,n. (30)

This is a master equation from which we can derive otherresults. The subscript n denotes that the value is individ-ual to a particular channel #n. For clarity, let us redefineβch so that given fLO and fave we always have

[fLOn + (1/2)βch,n] = fave,n. (31)

This way, the Eq. 30 will also be valid for asymmetricalbandshapes.

Let us suppose that all the channel delays are thesame within a given multiheterodyning unit, so thatτch,n = 〈τch〉 in Eq. 30. (We will allow them to bedifferent between units A and B, but same within A,and within B). This is reasonable if the delays are im-plemented by shift registers driven by a shared clock fre-quency. Let us describe the net delay as a deviation froma nominal delay τnet = 〈τch〉+∆τ . Then Eq. 30 becomes

∆φn = fLOn〈τch〉 + fave,n∆τ

= ∆φlin,n + ∆φdil,n. (32)

The first term is the phase shifts ∆φlin,n required to bringthe net device into phase-linearity, the second term is theadditional phase shifts ∆φdil,n required to change (dilate)the delay. Hence

∆φlin,n = fLOn〈τch〉 (33)∆φdil,n = fave,n∆τ. (34)

A. When Using a Pair of Devices Sharing fLO

Now suppose we have a pair of multiheterodyning units“A” & “B” that are being compared, such as to forma correlator or a double interferometer velocimeter, andwhich share the same reference frequencies fLOn. Sinceit is only a relative measurement, we need only changeone of the unit’s τnet, say unit B’s. The analysis belowassumes the channel delays and phases of unit A willremain the same, while we adjust the channel delays andphases of unit B. From Eq. 32 we have for the A and Bchannel phases

∆φn,A = fLOn〈τch〉A (35)∆φn,B = fLOn〈τch〉B + fave,n∆τ. (36)

Because a correlation or velocimetry measurement onlycares about a comparison between A and B within eachchannel, we are allowed to add an arbitrary constantConn to both A and B that could change from channelto channel.

∆φn,A = fLOn〈τch〉A + Conn (37)∆φn,B = fLOn〈τch〉B + fave,n∆τ + Conn. (38)

Now suppose we make the delays of A and B units thesame, 〈τch〉A = 〈τch〉B = 〈τch〉. This is fairly easy todo if the delays are implemented by shift registers drivenby the same clock frequency. Now Eq. 37 and Eq. 38proves that it is permissible to ignore the ∆φlin termsfor a correlator or velocimeter when the fLOn are sharedbetween the units, because that is equivalent to havingConn = −fLOn〈τch〉A so that

∆φn,A = 0 (39)∆φn,B = fave,n∆τ. (40)

Thus the simplest application of a correlator or velocime-ter sharing reference frequencies is to set all the channelphases to zero for both A and B units, and the channeldelays the same for A and B units. This is sufficient toadjust τnetB over a small range 〈τch〉 ± ∆τmax. For alarger range, we need to change both ∆τ and τch. Theanalysis below shows how.

Note that the net delay for this invention is more pre-cise than possible from a single channel used alone, be-cause the slope of the global line is based on the aver-age of many channel segments. Furthermore, if all thechannel delay values τch,n drift in the same direction,such as if they are shift registers driven by the sameclock frequency, then τnet does not change because that isequivalent to translating the global line vertically with-out changing its slope. Thus the temporal behavior ofthe multiheterodyning device is more precise and stablethan a single channel used alone.

In order to increase τnetB further than ∆τmax we mustincrease 〈τch〉B . In a shift register implemented channeldelay, τch will be naturally quantized in increments of aclock period, which we will call ∆τch. So let

〈τch〉A = 〈τch〉 (41)〈τch〉B = 〈τch〉 + k∆τch, (42)

where k is an integer which describes which delay rangewe are in. Then we could have

∆φn,A = 0 (43)∆φn,B = fLOn∆τchk + fave,n∆τ

= ∆φlin,n + ∆φdil,n (44)

for k = 0, 1, 2 . . . , and

τnetB = 〈τch〉 + k∆τch + ∆τ (45)τnetA = 〈τch〉. (46)

These four equations describe the configuration to usein a two unit correlator or velocimeter when referencefrequencies are being shared, and when some simplify-ing assumptions where used. For any circumstances, theexact result can be computed from the master equationEq. 30.

If the channel delays are not exactly the same betweenA and B units, this can be compensated for by adding a

4-27

phase adjustment to the B channels of (1/2)βch(τch,n,B−τch,n,A).

Fig. 8B shows a close-up of Fig. 8A where a channelsegment intersects a global line. The channel segmentis shown as a rectangle of vertical thickness 2∆φslop torepresent uncertainties in τch, ∆θ, and variation of ∆θwith intermediate frequency. Instead of assuming a con-stant ∆θ, the most accurate analysis of Fig. 8A wouldinclude the actual frequency dependence of ∆θ. Thiscould make the channel segment curved. For shift reg-ister delays driven by the same clock frequency, ∆φslop

could be insignificant compared to ∆φwalkoff .

B. Numerical Example of Choosing Phase Shifts

Let us give a numerical example using a 100 channelsof βch=200 MHz, filling an input bandwidth from 10 GHzto 30 GHz so that βnet=20 GHz. Each channel has a shiftregister delay driven by a clock frequency which must begreater than 400 MHz (according to the Nyquist crite-ria) in order for βch=200 MHz. Thus we can incrementthe shift register delay in quanta of ∆τch=1/(400 MHz)= 2.5 ns. Let the starting value of the shift register de-lay be τch = 2 ms = 2000000 ns. We can set ∆τ inEq. 44 to any value from zero to a maximum value of∆τmax ∼ (1/2)/βch ∼2.5 ns, which is when dephasingbegins. Note that conveniently, ∆τmax ≈ ∆τch, so thatjust as dephasing begins we can jump to a new delayrange by increasing τch,B by ∆τch, from 2000000 ns to2000002.5 ns etc.

We start with all shift register delays set to τch=2000000 ns, k = 0, and ∆τ = 0, and we set all the phaseshifts to zero according to Eq. 43 and Eq. 44. The net de-lay is the nominal value τnet= 2000000 ns. We are in thelowest delay range, and we can adjust τnet from 2000000ns to 2000002.5 ns by adjusting channel phases accordingto Eq. 44, by increasing ∆τ , up to ∆τ=2.5 ns. At thispoint we jump to a new delay range by incrementing k to1 and reseting ∆τ to zero. That is, now all the B-unitsshift register delays will be at 2000002.5 ns. The channelphases are nonzero because of the first term. The secondterm is zero. Now we can increase τnet from 2000002.5ns to 2000005 ns by increasing ∆τ from 0 to 2.5 ns andadjusting phases according to Eq. 44. And so on. Thusby the combination of changing the delay on a fine levelwith phases and on a coarse level by the shift registers,we can create any delay with very fine resolution.

C. Practical Advantages of Sharing ReferenceFrequencies

The invention is very practical because in some embod-iments it is not necessary to assign the reference frequen-cies fLO1, fLO2, fLO3, etc. to any precise set of values.They and their associated channel bandshapes can be ap-proximately scattered across the bandwidth βnet. This is

a great practical advantage because it allows the use ofinexpensive oscillators that may drift in frequency overtime. (The frequency must not drift significantly overthe delay time, but this is easily achieved.) Secondly, thedetailed shape of the bandshape does not critically affectthe autocorrelation or correlation in many embodiments(provided it is the same for all channels A, B, etc. sharinga given reference frequency). Thus inexpensive spectrom-eters and electrical components can be used. Thirdly, formany situations a variance in τch among channels is ac-ceptable because channel phase adjustment can compen-sate for it. This reduces cost of components creating thechannel delay.

D. Tolerance for Reference Frequency Drift

The tolerance to frequency drift for the reference fre-quencies can be estimated. The peak to peak spacing ofthe comb-filter is 1/τ . Suppose we want the drift in fre-quency to be less than 10% of a comb-filter peak-to-peakspacing (1/τ), and this drift occurs over a duration of τor the round trip time of the signal from apparatus totarget and back, whichever is greater. For radars, thelatter and former could be about the same, and be about2 ms. Thus we desire (10%)(1/[2 ms])=50 Hz stabilityover a 2 ms interval, or 25 kHz per second stability fora frequency that may be about 30 GHz. This is about 1part per million per second stability, which is achievable.

E. An Autocorrelator Sharing ReferenceFrequencies

Figure 18A shows an embodiment of the inventionforming a double interferometer velocimeter analogous toFig. 2A. The set of local oscillators 400, 401, 402, 403 etc.supply reference frequencies shared between two multi-heterodyning interferometer units “A” and “B” (406 and408). Phase compensators analogous to 510 in the cor-relator (Fig. 19A) are not necessary in this velocimetryapplication because the detecting interferometer (“B”)only cares about the power spectrum and is not phasesensitive.

The sharing of reference frequencies causes the ve-locimetry measurement to be insensitive to the choice ofspecific values for reference frequencies. Furthermore, thedetailed band shapes are not critical provided “A” and“B” units have similar shapes within each channel. Fig-ure 18B shows a hypothetical set of channel bandshapesfor one of the units, suggesting arbitrary placement andshape. The lack of commensuration between the comb-filter peaks 416 and 418 inside bands 412, 414 suggestslack of phase-linearity. Provided both “A” and “B” unitsdeviate from phase-linearity in the same manner, lack ofphase-linearity is acceptable. Also note that it is not crit-ical for the channel bands to contiguously fill the inputbandwidth βnet in this application, although it is optimal

4-28

to do so to produce the least ambiguous autocorrelationpeaks.

F. A Correlator Sharing Reference Frequencies

Figure 19A shows an embodiment of the invention per-forming correlation, where a set of reference frequen-cies fLOn are shared. This application correlates twowide bandwidth input signals 500, 502 for the purpose ofrange-finding, analogous to Fig. 2B. Each input signal isspectrally decomposed into channels and heterodyned bythe set of reference frequencies in the multiheterodynerunits 504, 506. When a set 508 of reference frequen-cies are shared, as shown in Fig. 19A, this creates thepractical advantage that the detailed value of the refer-ence frequencies is immaterial, analogous to the case withthe double interferometer velocimeter discussed above.Thus the bandshapes (for each channel pair A, B) andreference frequencies could be arbitrarily arranged. Ona coarse scale it is optimal to avoid overlapping chan-nel bandshapes, and optimal to fill the input bandwidthβnet contiguously. But on a detail level the particularfrequencies chosen do not matter.

Phase compensators 510 placed in series with the ref-erence signals are analogous to 795 in Fig. 17, and controldifferences in arrival time for reference signals from thesources 508 to each apparati 504, 506.

G. Synchronizing Widely Spaced Correlator Units

If the separate multiheterodyning apparati 504, 506 ofa correlator are spaced far apart, such as in radio astron-omy or long-baseline optical interferometry, then it maynot be practical to communicate the set of reference fre-quencies via a set of numerous cables. And even if thereference frequencies are combined into a single referencesignal, such as produced by a mode-locked oscillator, fre-quency dependent attenuation or dispersion may make itdifficult to communicate a wideband reference signal overlong distance. A solution is to use two sets of local os-cillators that are synchronized by cable 530, as shown inFig. 19C. The synchronization signal could have a narrowbandwidth to accommodate the best frequency for thelong distance cable. The synchronizing signal also couldbe expressed as an optical signal in the infrared whereoptical fibers have a minimum attention. The synchro-nizing signal could be used to synchronize two mode-lockoscillators, one at station 532 and one at 534, providingthe two sets of reference frequencies. Alternatively, thesynchronizing signal could be from an external sourcethat is accessible to both stations, such as a satellite orsimilar distant source.

H. Correlator for Optical Signals

Figure 5 shows a multiheterodyning correlator for op-tical signals “A” 702 & “B” 703. Reference signal 700 isadded to the input signal A (702) with a beamsplitter 704prior to the spectrometer 706, symbolized by the prism.The spectrometer could comprise a prism or a grating.The input angle of the reference beam into the prism orgrating may have to be slightly different than that of theinput beam so that the reference and beam fall upon thesame intensity detector element 708 when the referencefrequency is lower or higher than the portion of the inputsignal it will heterodyne. The intensity detector producesan electrical signal 712 in response to the difference fre-quency component between the input signal 710 and fLO

contained in reference beam 700. (The terms “power”and “intensity” can be interchanged because the area ofthe intensity detecting device is constant.)

Figure 6 shows a detail of the intensity detectors in thespectrometer of Fig. 5. The intensity detectors could beelements of a CCD detecting array. A column 740 of el-ements would be responsible for detecting the IF signalsand another column 741 responsible for the quadraturesignal *IF. Rows of detectors would be associated withspecific channel and with a specific range of wavelengths(frequencies). Let there be two polarization directions“h” and “v” also called horizontal and vertical. The in-put signal 702 is arranged to be polarized at 45 to “h”so that there is intensity in both the “h” and “v” com-ponents, and the reference signal is arranged to be cir-cularly polarized (or vice versa). A horizontal polarizerprecedes the IF intensity detectors and a vertical polar-izer before the *IF detectors. This way the *IF detectorssee a combined beam where the reference component lags90 behind relative to the IF beam. The DC componentof the intensity signal can be blocked by capacitor-likeelements 714, analogous to those labeled 757 in Fig. 7A.

Acknowledgments


4-29

V. CLAIMS

(Submitted, not official version)

1. A method for applying a signal process to a high bandwidthsignal to produce an output signal, comprising:

decomposing said high bandwidth signal into a plurality of spec-tral bands, wherein each spectral band of said plurality of spectralbands has a smaller bandwidth than said high bandwidth signal;

down-heterodyning each said spectral band to create an interme-diate frequency signal by beating each said spectral band againsta reference signal or against a component of said reference signal,wherein said reference signal comprises a plurality of reference fre-quency components, wherein each reference frequency componentof said plurality of reference frequency components comprises aunique frequency with respect to every other reference frequencycomponent, wherein for each said spectral band there is a specificreference frequency component having a frequency close to saidspectral band, wherein the set of said intermediate frequency sig-nals and their subsequent processing path is called a set of channels;

applying a signal process to each said intermediate frequencysignal to produce a set of channel output signals; and

summing said channel output signals to produce said outputsignal, wherein this method is called “multichannel heterodyning”or “multiheterodyning”.

2. The method of claim 1, further comprising up-heterodyningthe frequency of said channel output signals prior to the step ofsumming said channel output signals to produce said output signal.

3. The method of 2, wherein the step of up-heterodyning isperformed using the same said reference frequency components.

4. The method of claim 1, wherein the effective spectrum ofsaid signal process applied to said intermediate frequency signalis shifted in frequency by an amount individual to said channeland said amount can be expressed as a phase shift and is calleda channel phase shift, wherein said channel phase shift is appliedin a coordinated manner to many channels to modify the effectivetemporal behavior of said output signal.

5. The method of claim 4, wherein said coordinated mannerincludes having a component of said channel phase shift that causesthe effective temporal behavior of said output signal to change byamount ∆t, wherein said component is given by approximately theproduct of the average frequency of said spectral band associatedwith said channel times said amount ∆t.

6. The method of claim 4, wherein said coordinated manner in-cludes having a component of said channel phase shift that bringssaid output into phase-linearity, wherein said component for saidchannel is given by approximately the product of said reference sig-nal frequency component for said channel times an average over allsaid channels of the delay time characteristic of said signal processfor said channel.

7. The method of claim 4, wherein said channel phase shiftis accomplished by phase shifting said intermediate frequency sig-nal by an amount which is independent of frequency and called arotation angle to produce a phase shifted intermediate frequencysignal, wherein said phase shifted intermediate frequency signal iscalled a rotated signal, wherein said channel output is formed fromcombinations of undelayed and delayed replicas of said intermedi-ate signal having user-selected amplitudes, delay values and saidrotation angles, wherein said rotation angle is proportional to saidchannel phase shift and proportional to said delay values.

7B. The method of claim 7, wherein said intermediate frequencysignal is phase shifted by retarding said intermediate frequencysignal.

7C. The method of claim 7, wherein said intermediate frequencysignal is phase shifted by retarding said input signal in spectralband relative to said reference signal component.

8. The method of claim 4, wherein said channel phase shift isaccomplished by expressing said intermediate frequency signal as avector signal comprising said intermediate frequency signal and aquadrature manifestation of said intermediate signal, wherein said

quadrature manifestation of a signal is defined as that having aphase approximately 90 degrees different from said signal for all itscomponent frequencies, wherein said vector signal is rotated in itsvector space by an angular amount called a rotation angle; whereinsaid channel output is formed from combinations of undelayed anddelayed replicas of said vector signal having user-selected ampli-tudes, delay values and said rotation angles, wherein said rotationangle is proportional to said channel phase shift and proportionalto said delay values.

8B. The method of claim 8, wherein rotation of said vector is ac-complished by retarding said input signal in spectral band relativeto said reference frequency component.

9. The method of claim 8, wherein said vector rotation isproduced from a linear combination of said quadrature and non-quadrature vector components using coefficients which are sinu-soidal functions of said rotation angle in analogy to a mathematicalrotational transformation.

10. The method of claim 4, wherein said phase shift is achievedfor a given said channel by substituting for said intermediate fre-quency signal in said signal process said intermediate frequencysignal or quadrature version of said intermediate signal in differentcombinations to form a set of intermediate outputs, wherein saidchannel output is formed from linear combinations of said set of in-termediate outputs using coefficients which are sinusoidal functionsof an angular amount which is proportional to said phase shift.

11. The method of claim 4, wherein said input signal is anoptical wave, wherein said reference signal and said input signalin said spectral band comprise two beams, wherein one of said twobeams is circularly polarized relative to the other of said two beam,wherein both beams are detected by a first intensity detector sensi-tive to linear polarization, wherein both beams are passed througha first polarizer preceding said intensity detector, wherein said non-quadrature component is obtained by passing both beams througha second polarizer oriented orthogonal to said first polarizer.

12. The method of claim 1, wherein said signal process is anyprocess which can be Fourier decomposed and widely separatedfrequency components are independent of each other, wherein saidindependence means that the relationship between detailed phasesof said widely separated frequency components does not affect saidsignal process.

13. The method of claim 1, wherein said signal process is se-lected from a group consisting of a time delay process, a filteringprocess, an interferometry process, an autocorrelation process anda recording process, wherein said intermediate frequency signal iscalled an input signal and said channel output signal is called anoutput signal, wherein said output signal of said time delay processis a delayed replica of said input signal, wherein said output sig-nal of said filtering process comprises one or more delayed replicasof said input signal with an undelayed replica of said input signal,wherein said interferometry process is said filtering process, whereinsaid one or more delayed replicas of said input signal are optimallyspaced with the same interval, wherein said output signal of saidautocorrelation process includes time averaging the product of saidinput signal with a delayed replica of said input signal and includestime averaging the square of the output of said interferometry pro-cess, wherein said output signal of said recording process includesstorage and less than real-time manifestation of said input signalwhich may be in vector form, wherein the phase and amplitude ofsaid input signal is stored while simultaneously measuring the rel-ative phase and amplitudes of said multiple frequency componentsof said reference signal.

14. The method of claim 1, wherein said signal process is awaveform synthesis process, wherein said channel output signal isan ersatz form of said input signal, wherein said ersatz form in-cludes stored data, noise generators, and real-time signals havingbandwidth comparable to typical said intermediate frequency sig-nal.

15. The method of claim 1, wherein said input signal comprisesat least two separable signals which are to be processed by a mul-tiple input signal process, wherein said multiple input signal pro-

4-30

cess includes correlation, wherein said spectral decomposition andsaid down-heterodyning occur using separate apparatus to produceseparate intermediate frequency signals, wherein each said sepa-rate apparatus shares said reference signal frequency components,wherein said separate intermediate frequency signals are correlatedin a channel-by-channel parallel fashion organized by said referencefrequencies.

16. The method of claim 1, wherein said signal process is a cor-relation of at least two intermediate frequency signals, called inputsignals, derived from said spectral bands having similar averagefrequencies and said down-heterodyning performed using the samesaid reference signal component frequency, wherein a correlationfor two said input signals is based on the time averaged productof said two input signals, wherein a correlation for more than twoinput signals includes a user-defined function that maximizes whenall said input signals are replicas of each other, wherein a multipleinput correlation includes time averaging the square of the sum ofall said inputs.

17. The method of claim 1, wherein said signal process is acombination of signal processes selected from a group consisting ofinterferometry, filtering, autocorrelation, correlation, delay, record-ing, waveform synthesis, and arithmetic summing and subtractionof signals.

18. The method of claim 1, wherein the precise value of saidreference signal component frequencies may be arbitrary and unre-lated to each other, wherein it is optimal that the coarse value ofsaid reference signal components frequencies and average frequencyof said spectral bands be evenly positioned across the bandwidthof said input signal.

19. The method of claim 1, wherein said spectral bands aresparsely placed across the bandwidth of said input signal.

20. The method of claim 1, wherein said input signal includeselectromagnetic waves and sound.

21. The method of claim 20, wherein said input signal includesvisible and invisible light.

22. A method of comparing at least two wideband signals, com-prising:

passing each wideband signal of said at least two wideband sig-nals through a multiheterodyning unit, wherein reference frequencycomponents are shared between said multiheterodyning units.

23. The method of claim 22, wherein one of said multihetero-dyning units outputs a wide bandwidth illuminating wave havingat least one echo, wherein at least one of said multiheterodyningunits receives said illuminating wave,

24. The method of claim 23, wherein said at least one receiv-ing unit performs a signal process which includes correlation andautocorrelation.

25. The method of claim 22, wherein said multiheterodyningunits correlate said at least two wideband signals.

26. The method of claim 22, wherein said reference frequencycomponent sharing is achieved by synchronizing at least two sep-arate reference signal generators which create reference frequencycomponents for said multiheterodyning units.

27. A method of creating at least two matched wideband filters,comprising:

creating a filter with a multiheterodyning process using a firstset of reference frequencies,

creating a second filter with a multiheterodyning process using aset of reference frequencies which are the same as or synchronizedto said first set of reference frequencies.

28. The method of claim 4, wherein said input signal is an op-tical wave, wherein said reference signal and said input signal insaid spectral band comprise two beams, wherein for a first polariza-tion direction one of said two beams is retarded by a quarter waverelative the other said beam and relative to a second polarizationdirection orthogonal to said first polarization, wherein intensity ofsaid two beams in said first polarization direction is detected toform said intermediate frequency signal, wherein intensity of saidtwo beams in said second polarization direction is detected to formsaid quadrature intermediate frequency signal,

29. The method of claim 10.5, wherein one of said two beamsis circularly polarized relative to other beam of said two beams,wherein said other beam is linearly polarized at 45 degrees to saidfirst polarization direction.

30. The method of claim 14, wherein said signal process is thegeneration of a wideband signal having echos, wherein for eachsaid channel a waveform is generated which then passes throughan interferometer to form said channel output.

31. A method for producing an output signal, comprising:decomposing a high bandwidth signal into a plurality of spectral

bands;down-heterodyning each said spectral band of said plurality of

spectral bands to create an intermediate frequency;applying a signal process to each said intermediate frequency

signal to produce a set of channel output signals; and

summing said channel output signals to produce said output

signal.

IL-10000IL-10000M.tex 12/03

Part 5

Single and DoubleSuperimposingInterferometer Systems

US Patent 6,115,121Issued Sept. 5, 2000


Interferometers which can imprint a coherent delay on a broadband uncollimated beam are de-scribed. The delay value can be independent of incident ray angle, allowing interferometry usinguncollimated beams from common extended sources such as lamps and fiber bundles, and facilitat-ing Fourier Transform spectroscopy of wide angle sources. Pairs of such interferometers matched indelay and dispersion can measure velocity and communicate using ordinary lamps, wide diameteroptical fibers and arbitrary non-imaging paths, and not requiring a laser.

I. BACKGROUND OF THE INVENTION

A. Field of the Invention

The present invention relates to the design of interfer-ometers, and more specifically, it relates to the design ofsingle interferometers and pairs of interferometers whichcan use broadband uncollimated light from an extendedsource.


Interferometers in single and in double arrangementsare very useful devices in a variety of applications inoptics, metrology and communications. These interfer-ometers have a delay τ between their arms which canproduce fringes in the output intensity. Single inter-ferometers having zero or near zero delay can be usedwith short coherence illumination to compare distancesbetween one arm containing a sample and a referencearm. Single interferometers having larger delays can beused to measure the spectrum of a source, or to mea-sure Doppler shifts of laser illuminated targets to mea-sure motion, since slight differences in wavelength (λ)cause changes in fringe phase. Double interferometershaving non-zero delays are useful in optical communica-tions and measurement of velocity of targets illuminatedby broadband radiation.


1. Superimposing Interferometers and Uncollimated Light

Prior art interferometers have a design which requiresparallel light, or which is not optimized for very broad-band usage, or was impractical for very long delays. Con-sequently, these interferometers are impractical for manyapplications involving white light and light from an ex-tended source, such as from a common lamp, which pro-duces non-parallel (uncollimated) beams. This is a seri-ous handicap, because the majority of inexpensive com-pact sources are then excluded.

The present invention includes the use of interfer-ometers, in single or in pairs, which are capable ofworking with broadband uncollimated light. These arecalled achromatic “superimposing“ interferometers be-cause they superimpose the images or ray paths asso-ciated with each interferometer arm. A similar concepthas been discussed previously by others and called “fieldcompensation.” However, previous interferometer de-signs may have been superimposing only for a very lim-ited wavelength range, or are impractical to create largedelays, or are impractical to scan the delay.

2. Coherent Delays

Generally, the goal in a superimposing interferome-ter is to split an input beam into two or more separatebeams, coherently delay one beam relative to the other,and recombined them to form an output beam. (Usuallyjust two beams are split and recombined in the case of aMichelson-type interferometer, or an infinite number inthe case of a recirculating Fabry-Perot type interferome-ter). The separate, splitted beams are also called interfer-ometer “arms,” and the arms can have different lengths

5-2

A Coherent Delay

Uniform delay

Uniform delay

Fig. 1A

Input image plane

Output image plane

40

44

46

48

50

42

Fig. 1B

High qualitySuperimposingInterfer.

Inputimageplane

Delay

Outputimageplane

58

56

60

Superimposed inpath and image

Fig. 1C

Lesser qualitySuperimposingInterfer.

Inputimageplane

Outputimageplane

58

52

54

Superimposed inimage, not in path

Delay

Figure 1A shows a coherent delay between pixels of input andoutput planes. Figure 1B shows the echo and undelayed signalsuperimposed in path (and also image). Figure 1C shows theecho and undelayed signal superimposed in image but notpath.

which create a relative delay. For each input pulse, twoor more pulses are outputted. The earliest output pulse,from the shortest arm, could be called the “undelayed”signal. The later pulses, from the longer arms, are called“echos.” The interval between the undelayed signal andthe echo is the interferometer delay τ .

The term “coherently delayed” is explained in Fig. 1A.A pattern of light (which could be called an image) maybe presented to the input image plane 40, and the in-terferometer transports the pattern to an output plane42 by imaging optics, so that there is a correspondencebetween input and output pixels 46 and 50, and between44 and 48. The pattern appears for each instance of theoutput pulses discussed above. We are interested in thedelay between instances. The term “coherently” delayedmeans that the time delay τ1 for rays of light arriving atoutput pixel 50 is the same for all the rays of the bundlearriving at 50, within a tolerance of a quarter wave (atime interval of λ/4c, where c is the speed of the wave).The delay associated with another output pixel 48 couldbe called τ2, and this may be different or the same as τ1.Thus, even though the delay τ may change from output

pixel to pixel, within the bundle of rays associated witheach pixel the delay is very uniform. This uniformityis necessary to create fringes having significant contrast(visibility).

In some applications it is desired that τ have the samevalue for all pixels of the input beam. This will pro-duce an infinitely wide fringe on output of the interfer-ometer. In other applications it may be desired that τvary linearly across the output beam. This will gener-ate an evenly spaced and parallel fringe “comb,” wherethe phase of the fringe varies linearly transversely acrossthe beam. (This is usually easily accomplished by tiltingan interferometer arm end mirror.) Only superimpos-ing interferometers can produce infinitely wide or evenlyspaced parallel fringe combs. (Conventional Michelsoninterferometers produce rings of fringes which are notlinear or evenly spaced, except very far off axis). A su-perimposing interferometer could also be called angle-independent or solid-angle independent, because τ is in-dependent of the angle of each ray within the bundleassociated with a given image pixel.

A note on terminology: the delay value can be specifiedseveral ways, by single-trip path length D, by the roundtrip path length cτ , or by a time interval τ . Strictlyspeaking, the units of τ should be time, but for conve-nience τ is sometimes specified in terms of length units,in which case it is implied that the length should be firstdivided by c to yield the equivalent time value. Secondly,these delay values usually refer to the difference in pathlengths or travel times between interferometer arms, ex-cept when an isolated means for creating a delay is dis-cussed, in which case it could refer to the propagationtime through this means. When an optical element orassembly is inserted into an optical path, the increase inthe propagation time τ causes over the original time iscalled an “insertion” delay. For example, a 20 mm thickglass etalon having a refractive index of n=1.5 would havean insertion delay of ∆D = (n − 1)20 = 10 mm, due tothe slower propagation of light through glass comparedto vacuum or air. Thirdly, the term “delay” has dualmeanings in this document. It may refer to a value, suchas τ , cτ or D, or it may refer to the means for creatingthe delay value, such as “an etalon delay” or a “relay lensdelay”.

3. Angle Dependence of Classic Michelson and Fabry-PerotInterferometers

Figure 2A shows a common Michelson consisting of abeamsplitter 24 and two end mirrors 20 and 22. Thisinterferometer is not superimposing because mirrors 20and 22 do not superimpose in view of the beamsplitter.This has the consequence that τ depends on incident rayangle. The single-trip path lengths between the beam-splitter and end mirrors differ by an amount D for raysparallel to the optic axis. This causes a round trip de-lay time of τ = 2D/c between the two arms, where c is

5-3

D=1/2 c

Fig. 2A

24 20

22

Figure 2A is a conventional Michelson interferometer.

Effective D

D

20 22Fig. 2B

25

10

11

Fig. 2C

D

21 23 2627 28

Figure 2B is the optical equivalent of Fig. 2A and shows theangle dependence of delay and lack of superposition of out-put paths. Figure 2C shows the lack of superposition for aconventional Fabry-Perot interferometer.

the speed of the illumination (which is spoken of as light,but can be any radiation that travels as rays, such asmicrowaves, sound, x-rays). Figure 2B shows the opti-cal equivalent of Fig. 2A, because the partial reflectionoff the beamsplitting surface 24 puts the two mirrors 20and 22 apparently in the same path, but longitudinallydisplaced. When a ray of light 25 enters with an angleθ to the optic axis, it encounters a path difference whichdepends on angle as 2D cos θ. Thus the delay deviatesfrom its nominal value of τ by τ(1 − cos θ), which is ofthe order (1/2)τθ2, for small angles with θ in radians.This deviation can smear the delay which reduces fringevisibility. Note also that the ray 10 reflecting off mirror20 does not overlap the path of ray 11 from the other mir-ror 22. For the same reason, an object seen in reflectionof the two mirrors 20 and 22 will be seen as two imagesthat are longitudinally displaced. Thus, the Michelsonhaving a nonzero delay does not superimpose ray pathsnor images, except for perfectly parallel incident light.

The non-superposition and the angle-dependence of theinterferometer are related.

The common Fabry-Perot interferometer (Fig. 2C)consisting of two flat partially reflecting mirrors 21, 23separated by a distance D, is non-superimposing andsuffers an angle dependence to the delay analogous tothe Michelson example. The output rays 26, 27, 28for a single given input ray 29 do not superimpose inpath. Furthermore, the multiple images of an object ob-served through the interferometer will be longitudinallydisplaced from each other. Thus this interferometer doesnot superimpose paths nor images.

The angle dependence of these non-superimposing in-terferometers creates practical difficulties, because thefringe visibility is small unless the incident light is veryparallel. An illumination source producing rays havinga range of cone angles up to θ will smear the delay by∼ (1/2)τθ2, which destroys the fringe visibility if this ismore than a quarter wave (λ/c4). This puts a limit of

θ ∼√

(λ/2cτ) (1)

for the maximum cone angle. In order to produce thisdegree of parallelism from an ordinary lamp, which is anextended source, a small pinhole must be used a far dis-tance from the interferometer. This greatly reduces theamount of power available from the lamp. For example,for a 4-meter delay (such as used to measure Doppler ve-locities typical of automobiles) and green light (λ=500nm), θ must be less than 0.00025 radian. This limits thenumerical aperture of the illumination source to f/2000,which greatly reduces the amount of power available froma filament. Secondly, this restriction on parallelism alsoapplies to the reflected light from the target as it entersthe detecting interferometer in a velocity interferometerapplication. This severely reduces the depth of field ofthe target motion, so that when the optics imaging thetarget into the interferometer become slightly out of fo-cus, visibility of the fringes is diminished.

Single interferometers (Fabry-Perot and Michelson)are used as for spectroscopy, such as Fourier Trans-form spectroscopy. The small cone-angle tolerance ofthese non-superimposing interferometers limits the spec-tral resolution. A book by R. Beer shows that for con-ventional Michelson interferometers there is a reciprocalrelationship between the solid angle Ω = πθ2 of a sourceand the best spectral resolution (λ/∆λ) achieved with asingle channel detector, so the higher the spectral res-olution the smaller the signal power. (Reinhard Beer,“Remote Sensing by Fourier Transform Spectrometry”,John Wiley & sons, NY 1992, QD96.F68B43, page 17).The limit on Ω for a given spectral resolution severelylimits the etendue or light gathering power (beam areatimes Ω), and prevents high spectral resolution on diffusesources such as the aurora, plasmas, light from specklingimages of stars, or light communicated through large di-ameter optical fibers.

In contrast, with a superimposing interferometer thedelay is independent of ray angle and all the light can

5-4

Fig. 1D

62

6468

6667

69

Input/output plane65

Figure 1D is an example interferometer that superimposesimages but not ray paths.

be accepted from an extended source and have the samedelay imprinted on it coherently. This can dramaticallyincrease signal power. For the above example where thenumerical aperture of a conventional Michelson was lim-ited to f/2000, for a superimposing Michelson the nu-merical aperture could be, say f/10. It is not limited byanything having to do with the delay time, but instead bythe diameter of the optics used in its construction. Theamount of light power is increased by the ratio (2000/10)2or 40,000. This is a tremendous advantage.

4. Superposition Creates Angle Independence

A method of making an angle-independent delay is tosuperimpose the ray paths (Fig. 1B) associated with eachinterferometer arm. This is the best solution. A less de-sirable method, but still useful, is to superimpose imagesassociated with each interferometer arm (Fig. 1C). Fig-ure 1B shows the rays for a single pulse 58 that enters apixel at the input image plane will appear at a pixel atthe output plane, and there will be at least one echo 56 tothe main signal 60, and that the ray paths for the echossuperimpose the paths of the main signal. In Fig. 1C therays for the echos 52 and signal 54 do not share the samepath, but still intersect at the output pixel. Note thatsuperimposing paths automatically superimposes images,but not vice versa. Superimposing paths is preferred be-cause then the detector can be placed anywhere along theoutput optical path. If only images are superimposed,then the detector must be placed at the output plane ora re-image of this plane, otherwise the rays from all thearms do not intersect properly at the appropriate pixelsand the fringe visibility is poor. Devices that are ide-ally designed to superimpose paths may in practice haveslight aberrations that cause the paths of one arm to de-viate from the intended path, so that strictly speakingonly a superposition of images is achieved. Thus there isnot a black-and-white distinction between the two kinds,it is a matter of degree.

Figure 1D shows an example, using a Michelson in-terferometer having zero delay. The interferometer has aplane mirror 62, and an irregular mirror 64 superimposed

foci

Fig. 3A

D

3430

35

33

32

Optic axis

31

foci

Fig. 3B

∆D

36

3738

Figure 3A shows a Spherical Fabry-Perot interferometer. Fig-ure 3B shows the effect of changing the mirror separation ofa Spherical Fabry-Perot interferometer.

longitudinally in reflection of the beamsplitting surface68. Let both the input and output planes 65 be at thesemirrors. Then the irregular surface of mirror 64 will causethe output ray 67 to have a different angle and hence paththan output ray 66 from the flat mirror, yet both rays ap-pear to come from the same pixel 69 of the output plane.Thus images are superimposed while paths are not. Inthese cases, it is very important that the detector be atthe output image plane or a re-image of that plane. Thisdiscussion is meant to illustrate the utility of defining in-put and output planes for realistic interferometers, thatis, those having slight aberrations.

5. Superimposing Fabry-Perot

The ray path superposition principle has been dis-cussed in the design of the spherical Fabry-Perot(Fig. 3A) [Pierre Connes, “L’ Etalon de Fabry-PerotSpherique”, Le Journal De Physique et le Radium 19,p262-269 (1958)], and the wide-angle Michelson interfer-ometer [R.L. Hilliard and G.G. Shepherd, “Wide-angleMichelson Interferometer for Measuring Doppler LineWidths”, J. Opt. Soc. Am. 56, p362-369 (1965)], whereit was called “field compensation”. These interferometerdesigns have properties that discourage or prevent theiruse in applications where broadband illumination is used,or long delays are needed, or adjustability of delay is de-sired.

Figure 3A shows a Spherical Fabry-Perot, consistingof two spherical mirrors spaced such that the two mirrorfocal points 30 coincide in the middle of the distance sep-arating the mirrors. Each mirror has a half 32 which istotally reflective, and a half 33 which is partially reflec-tive. A ray 34 entering the cavity recirculates between

5-5

the left and right mirrors, emitting a series of outputpulses with geometrically decreasing intensities. The in-terval between output pulses is called the interferometerdelay τ . Because of the overlap of foci, τ is independentof input ray path and the output rays for a given inputray 34 superimpose in output path 35.

The bottom half 32 of each mirror must be totally re-flecting to allow only rays that have made an even numberof round trips between the left/right halves to emerge. Ifboth halves (32, 33) of the mirror were partially trans-parent, then output rays would be emitted which wouldnot superimpose with rays 35. Another way to thinkabout it is that if both halves were partially transparent,then an image plus an upside down version of that imagewould be outputted. The presence of the upside imagewould spoil the fringe visibility. Note that the edge be-tween the totally 32 and partially reflective halves 33 lieson the optic axis 31. This is inconvenient because it pre-vents full use of the circular region of the output imagearound the optic axis where aberrations, such as spher-ical aberration, are smallest. The spherical Fabry-Perotmust therefore be used slightly off-axis.

For the Spherical Fabry-Perot it is not possible to ad-just τ and maintain the superimposing condition becausethe focal lengths are fixed. Figure 3B shows that whenthe separation of the mirrors is changed, the two mirrorfoci 36 come out of overlap. Then the undelayed 37 andfirst echo output ray 38 for a given input ray no longer su-perimpose and the delay τ becomes dependent on inputray path.

6. Virtual Imaging by an Etalon

The wide-angle Michelson discussed by R.L. Hilliardachieves path superposition for monochromatic light byuse of a glass etalon, as shown in Fig. 4A. Due to refrac-tion through the glass etalon 76, the mirror 72 behindthe etalon appears at 74, closer to the beamsplitter 71by a displacement B given by T (n − 1)/n, where T isthe etalon thickness and n is the refractive index of theetalon. Mirror 70 of the other arm is arranged to super-impose with the apparent mirror position 74. This cre-ates a time delay between the two arms due to the sum oftwo effects: the actual mirrors are at different distancesfrom the beamsplitter, and secondly, light travels slowerthrough glass by the factor n. The net delay is

cτ = 2[T (n − 1)/n + T (n − 1)]. (2)

Due to glass dispersion the apparent mirror position74 is different for different wavelengths. This preventsthe superposition condition to be achieved for all thewavelengths of white light, so that this interferometeris inappropriate for a white light velocity interferometer.Secondly, long delays such as cτ = 4 meters, are not prac-tical. (These long delays are useful for measuring me-ter/sec velocities found in industry). Thirdly, the delay

Glass etalon

7674 72

70

71

BT

Fig. 4A

77 75

73

Delaying mirror

Fig. 4B

Source or object

Etalon

Apparent location

Beamshortening

Physical length

Apparent length

966964

960

962

968

970

972

Figure 4A is a Michelson using an etalon to achieve superpo-sition. Figure 4B shows what is meant by the term “beamshortening”.

value cannot be adjusted by a significant amount. (Tilt-ing the etalon adjusts it slightly, but introduces astigma-tism).

These disadvantages are not important when theetalon Michelson (Fig. 4A) is used as a velocity inter-ferometer with monochromatic laser illumination. How-ever, there is great utility in being able to use cheapordinary white light sources for illumination instead ofan expensive laser, and these call for a different kind ofinterferometer.

7. Double Interferometers for White Light Velocimetry

Recently, an interferometric method of using broad-band illumination to measure target motion has been in-vented by the present author. The theory of operation isdescribed in patent 5,642,194 by David J. Erskine, whichis included herein by reference. In concept, it uses twointerferometers in series, with the target interposed. Inorder to use white light, and uncollimated light from ex-tended sources, the interferometers must be superimpos-ing for a wide range of wavelengths. Furthermore, inorder to measure slow velocities, of order meter/second,long delays of several meters in length are needed. Inthis white light velocity interferometer, the two interfer-ometer delays must match. This requires adjustability of

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White lightDelay

1

Delay2

1

3

5

7

9

FIG. 5A

Illuminatinginterferometer

Detectinginterferometer

8

Figure 5A is a topological diagram of a double interferometerconfiguration able to measure velocity.

the second interferometer delay. All these requirementshinder practical use of the prior art interferometers in thewhite light interferometer, or in other double interferom-eter applications, such as communication, where incoher-ent lamps or multimode fibers are desired for illuminationsources.

The topology of a double superimposing interferometerconfiguration used for broadband velocimetry is shown inFig. 5A. (The configuration for optical communication issimilar, except the target 5 is omitted.) Two superim-posing interferometers (3 & 7) are in series with an in-terposed target 5 and illuminated by a broadband source1. When the delays τ1 and τ2 match within a coherencelength of the source illumination, fringes are created inthe intensity of the outputs 8, 9, which are complemen-tary in phase. In some embodiments, a single actualinterferometer is used to implement the first and secondconceptual interferometers by using retro-reflected lightfrom the target. This automatically matches τ1 = τ2, inspite of gross mechanical vibration of the instrument thatmay change τ . This makes the retro-reflecting configura-tion attractive for industrial environments, and reducesweight and cost of the optical platform, since this doesnot have to be as rigid.

Target displacement or refractive index along theround trip path to the target during the interval τ causea proportional shift ∆φ in the fringe phase, where φ isin units of fringes. (One fringe is 1 revolution, 360 or2π radians.) Thus, this is essentially a radial velocitymeasurement, as opposed to some other systems whichmeasure transverse velocity (such as those involving theintersection of two incident laser beams to create stand-ing waves, through which a particle to be measured trav-els). However, the combination of several simultaneousradial velocity measurements taken at different angles tothe target can provide all 3 components of the velocityvector if desired.

When the light reflects normally off the target so thatit approximately doubles back on itself, the displacement∆x during an interval τ is

∆x = (λ/2)∆φ (3)

and the average velocity v over that interval is

v = (λ/2τ)∆φ (4)

so that the velocity per fringe proportionality is (λ/2τ).The phase shift produced by a moving target is ∆φ =v(2τ/λ). Hence, in order to produce a significant fringeshift to detect a slow moving target, long delays are de-sired. A cτ = 4 meter delay produces a fringe shift pro-portionality of ∼20 m/s per fringe. Since fractional fringeshifts down to λ/100 can be easily measured, a 4-meterdelay can have a velocity resolution of 0.2 m/s, suitablefor industrial settings. Now the challenge is to build a 4-meter superimposing interferometer that is achromatic.Clearly, a 4-meter thick glass etalon is not practical dueto its cost and chromatic aberration. Some of the designspresented below are a solution to this challenge.

8. Broad Bandwidth Illumination

There are different ways of detecting and interpretingthe interferometer output. The output light can be de-tected by a single detector that is sensitive to a widebandwidth, in which case an average λ is used. Alterna-tively, the output can be spectrally resolved into multiplechannels. In this case, the velocity can be computed foreach channel using the λ specific to each channel. Thesewill give redundant velocity answers which can be usedto check for consistency.

One advantage of using wide bandwidth illuminationis the unambiguous velocity determination. That is, thelack of integer fringe skips when the velocity jumps morerapidly than the detecting electronics can follow. Essen-tially, the velocity measurement taken in different colors,such as red, green, blue, can be combined to determinethe absolute velocity unambiguously, even though indi-vidually each color channel may have an integer fringeambiguity. This is because each channel has a differentvelocity per fringe proportionality.

In contrast, in the conventional monochromatically il-luminated systems, the fringe phase is ambiguous to aninteger due to the periodicity of monochromatic fringes.This creates a great uncertainty in the gross value of thevelocity when the target is first acquired (such as a carcoming over the horizon) or if the signal drops out tem-porarily. This fringe skip uncertainty hinders the use ofthese monochromatic systems in applications where thereis no other confirming method of velocity measurement,or where theoretical prediction of velocity is poor.

9. Velocimetry with a Double Fabry-Perot

A similar velocimetry method using two Fabry-Perotinterferometers and laser illumination was described ina journal article by S. Gidot and G. Behar, “Multiple-line laser Doppler velocimetry”, Appl. Opt. 27, p2316-

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2319 (1988). However, since the Fabry-Perot is not a su-perimposing interferometer, this method is not practicalwith uncollimated illumination. Uncollimated rays pass-ing through the Fabry-Perot will blur the delay value,which can cause a loss of fringe visibility. This producesa severe trade-off between degree of non-parallelism andmaximum source bandwidth which precludes practicaluse of common white light sources such as lamps.

10. Beam Shortening

Figure 4B shows what is meant by the term “beamshortening”, which is an important part of creating a su-perimposing interferometer. When real or virtual imag-ing by an optical element 966 causes an object or source960 to appear by ray tracing to be at a different location962 than the actual physical location 960, then the dif-ference 964 is called a beam shortening or beam lengthshortening. The beam length could be measured fromsome reference plane 968. The apparent beam length970 is from plane 968 to the apparent object location962. The physical beam length 972 is from plane 968 tothe physical object location 960. Figure 4B shows a pos-itive beam shortening, when the apparent beam lengthis less than the physical beam length. Negative beamshortenings are also possible.

In those interferometers where the beam reflects off anend mirror, such as 72 in Fig. 4A, and nearly doublesback on itself, the beam shortening changes the positionof the end mirror 72 from its physical position 72 to anapparent position 74. The roundtrip beam shorteningwould be twice B. When beam shortening involves anend mirror, the combination of the end mirror and theoptical element performing the imaging could be called a“delaying mirror”. In Fig. 4A, the delaying mirror wouldbe etalon 76 and end mirror 72. The term “delayingmirror” replaces the term “superimposing delay” used inthe previously mentioned patent 5,642,94 by the presentauthor.

A delaying mirror is a set of optics that acts like amirror in terms of ray tracing, but delays the waves intime compared to an actual mirror. Delaying mirrors areuseful elements in forming a superimposing interferome-ter. This document present designs of delaying mirrorsand designs of superimposing interferometers that maybe useful because of their achromatic character, theirpossible long delay times, adjustability, or wide imagefield.


A superimposing interferometer involves beam short-ening so that two paths appear to have the same lengthby ray tracing, but may have different propagation times.Many interferometers involve beams that reflect off anend mirror, so that the path nearly doubles back on it-

self. When beam shortening is applied to such configu-rations, a virtual mirror is formed that appears to be ina different location than its actual location. This couldbe called a “delaying mirror”.

A delaying mirror is a name for an apparent mirror(A.M.) created by real or virtual imaging of an actualmirror, called an end mirror. The rays incident the A.M.surface leave the surface in a direction as if they reflectedfrom an actual mirror located at the A.M. surface, exceptthey are delayed in time compared to if an actual mir-ror were located at the A.M. surface. The delay timeis controlled in part by the distance between the A.M.and the end mirror, and the amount of glass thicknessused in the optics that performs the imaging. The A.M.can be curved or flat, but is usually flat. The position ofthe A.M. is calculated by imaging the surface of the endmirror. The center of curvature of the A.M. is calculatedby imaging the center of curvature of the end mirror.The A.M. is optimally independent of illumination wave-length. Delaying mirror kinds can be classified by themethod of imaging used, real or virtual, the arrangementand number of elements performing the imaging, and thekind of end mirror.

A superimposing interferometer is a kind of interferom-eter that superimposes the path of each of at least twooutput rays produced by a given input ray. There maybe a delay between each of these output rays, and thisdelay could be zero. The superimposing interferometerhas the desirable property that the delay is independentof incident ray angle, for a given location of the ray whereit intersects a so-called input plane. (This location is alsocalled a pixel of an input image.) This allows a coherentdelay to be imprinted on an uncollimated beam, and al-lows the use of extended illumination sources in interfer-ometry. The superimposing interferometer is most usefulwhen the superposition is maintained over a range of il-lumination wavelengths (and thus is called achromatic),so that interferometry may be performed with uncolli-mated beams of white light, from ordinary lamps thatare extended sources.

If the exact superposition of path is not achieved, alesser, but still useful, kind of superimposing interferom-eter can be formed by requiring that the at least two out-put rays associated with each input ray superimpose inlocation at a so-called output plane. This could be calledsuperposition of images, where each interferometer armhas an image associated with it. The images must su-perimpose longitudinally, transversely, in magnification.(Note that the superposition of path is a more stringentrequirement equivalent to superposition of images whilealso matching wavefront curvatures.) By placing the de-tector that records the fringes at the output plane orsome relayed image of it, fringes with significant visibil-ity are still observed. An interferometer that satisfies thesuperposition of paths is more useful than one that sat-isfies only superposition of images because when pathsare superimposed, the detector can be placed anywhereto record visible fringes.

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Superimposing interferometers can be formed by su-perimposing one or more mirror-like elements by the par-tial reflection of surfaces such as a beamsplitters. Themirror-like elements may be actual mirrors or apparentmirrors (delaying mirrors). The interferometer kinds canbe classified by how the superposition is configured, andby the kind of delaying mirror involved. If the mirrorsface generally in the same direction as seen in the beam-splitters, then a finite number of output pulses is emittedfor each pulse incident on the interferometer, and an N-path interferometer is formed. A superimposing Michel-son is a kind of 2-path interferometer. If the mirrorsface each other so that a beam bounces back and forthin a recirculating path, then an infinite number of geo-metrically decreasing intensity pulses is emitted for eachinput pulse. Then the interferometer is of the recirculat-ing class. A Fabry-Perot interferometer is a member ofthis class.

Superimposing interferometers can be formed withoutdelaying mirrors, by explicitly routing the beams so thatthere are more than one path between input and outputplanes, and by requiring that for each path the inputand output planes are superimposed. Both N-path andrecirculating classes (kinds) can be formed this way.

Single superimposing interferometers are useful forspectroscopy, especially from extended or wide anglesources. Series pairs of superimposing interferometerswith matched delays and dispersions are useful for veloc-ity interferometry and communication using illuminationwhich can be broadbanded and extended, such as fromcommon lamps. A procedure for matching the delaysand dispersion between interferometers while individu-ally maintaining the superposition condition for each in-terferometer involves iteration. The superimposing con-dition is optimized for one interferometer by temporar-ily using the interferometer in a retro-reflective mode byobserving a retro-reflective target. The second interfer-ometer is then matched in delay, and then in dispersionwhile holding the delay constant.

In some applications, having a coherent delay whichvaries across the output plane is useful, since this mea-sure a range of delays in a single moment. This is calledan inclined delay. Tilting a mirror is a method of achiev-ing an inclined delay. However, only small values of in-clination are practical by this method before the fringevisibility diminishes. A large inclination can be achievedby the use of wedges. Furthermore, by use of more thanone wedge having different dispersive powers, an achro-matic inclined delay can be achieved. In an analogousfashion, achromatic delays using etalons of different dis-persive powers can be achieved. This removes a previouslimitation of etalon delays in broadband interferometry.

When a single interferometer is used to observe as wellas illuminate the target, this arrangement is called usingthe interferometer in a retro-reflective mode. The advan-tage is that the delay and dispersion matching conditionsare automatically approximately satisfied, in spite of se-vere change in the delay such as due to vibration of the

mounts holding the optics. However, in order to use thisconfiguration, a method of preventing illuminating lightfrom reaching the detector before reflecting off the targetmust be found. Several methods are described, includinghaving the beams travel at an angle to the optic axis,offsetting the images of the target for incoming and out-going beams, and using orthogonal polarizations.

A superimposing interferometer can be constructedthat outputs more than two pulses for each input pulse,so there could be two or more delay values between thefirst output pulse and its echos. This would be an N-pathinterferometer with N greater than two. When the delaysvalues are different, this is useful for generating simulta-neous fringe patterns characterized by different delays.This is useful, for example, in velocimetry for reducingvelocity ambiguity and in spectroscopy for measuring dif-ferent portions of a Fourier Transform fringe record si-multaneously, to make the measurement more robust toa fluctuating source. The different delays could also bedistinguished by polarization by incorporating polariza-tion elements in the interferometer.

A superimposing interferometer can be combined inseries with a prism or diffraction grating to make a spec-trum that has uniformly spaced fringes on it, with anadjustable spacing. This is useful, for example, for in-creasing the fringe visibility in Fourier Transform spec-troscopy or velocity interferometry.

Methods of using a delaying mirror to coherently delaya beam are shown, where it is desirable to separate theincoming from outgoing beams.


A superimposing interferometer can be made by super-imposing a delaying mirror with an ordinary mirror, orwith another delaying mirror. The different kinds of in-terferometers can be classified by how the superpositionis configured, and by how the delaying mirror is formed.If the delaying mirror and other mirror are superimposedfacing the same direction, such as by using partial reflec-tion/transmission from a beamsplitting cube, then theinterferometer formed is a kind of Michelson interferom-eter. If the delaying mirror and other mirror face eachother so that light is reflected back and forth, then a re-circulating interferometer is formed, similar to the Fabry-Perot (F-P).

A delaying mirror is a set of optics that acts like amirror in terms of ray tracing, but delays the waves intime compared to an actual mirror located at the sameplace. In Fig. 6A the delaying mirror assembly is the setof optics 82, 84 and 86 that act as a mirror for one armof a Michelson class interferometer. The “business” endof the delaying mirror assembly is the apparent mirror(A.M.) surface 80. This is where the delaying mirrorappears to be. In general, the A.M. is created by realor virtual imaging of an actual mirror 86, which is called

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A.M.

90

88

80

82 84 86

Field Center End

Fig. 6A

S.M.

Delaying Mirror Assembly

Figure 6A shows a superimposing interferometer using a relaylens delay with a spherical end mirror.

the end mirror, located a distance away. In Fig. 6A realimaging is used, and this is accomplished by the centerlens 84. The field lens 82 is used to flatten the A.M.surface, which would otherwise be curved. The delaytime of the delaying mirror assembly is controlled by thedistance between the A.M. and the end mirror and thenet amount of glass along the optic axis.

Michelson-class interferometers usually are consideredto have just two arms. The arm containing the delayingmirror could be called the long arm, and the other armthe short arm. The mirror 88 could be called the shortarm mirror (S.M). In general, the S.M. 88 could be anactual mirror, as in Fig. 6A, or a second delaying mirrorassembly, as will be discussed in the section on differentialinterferometers.

To create the superimposing interferometer, the A.M.surface is superimposed with the S.M. This creates aninterferometer that for ray tracing purposes acts like azero-delay interferometer, and is therefore independentof ray angle. By time-of-flight however, there is a non-zero delay time because it takes a finite amount of timefor light to pass through the delaying mirror assemblyand leave out the A.M. Thus, the spectral properties of anon-zero delay interferometer are achieved while havingangle independence.

Optimally, the overlap between the A.M. and themirror-like elements of the other arms is as perfect aspossible everywhere across the A.M. The greater the de-viation of the A.M. from the S.M., the greater the lossof fringe visibility for uncollimated beams. Since usuallythe S.M. is a flat mirror, usually the A.M. should be flat.When the S.M. is a curved mirror, the A.M. should becurved with the same radii of curvature. For concrete-ness, we will generally assume the S.M. is flat and thusa flat A.M. is desired.

If the A.M. surface is perfectly superimposed withthe S.M. surface, then superposition of paths has beenachieved (as in Fig. 1B), which is ideal. In this case, theinput and output image planes for the interferometer canbe anywhere. However, due to aberrations of the optics,the A.M. surface curvature may not perfectly match theS.M. curvature. In this case, it is possible to overlap thelocation of the A.M and S.M. for some sections of the

A.M.

90

88

80

82 85

86

Fig. 6B

Field Center

End

S.M.

Figure 6B shows the center lens of the relay delay imple-mented by a spherical mirror.

A.M.

Fig. 6C

80

91 92 9390

95

88

90

Field1 Center Field2End

S.M.

Figure 6C shows the relay delay implemented by transmissivelenses and a plane end mirror.

beam, while the angles of the surfaces are slightly dif-ferent. This superimposes images but not paths (as inFig. 1C). This can still produce visible fringes providedthe detecting device is placed at a relayed image of theA.M. surface. That is, the input and output image planesshould be located at the A.M. (This assumes that thebeamsplitting surface is flat. Generally, the input andoutput planes should be optimally placed closest to theoptics that generates the most wavefront irregularities,whether this be the beamsplitter or A.M.).

A description of different interferometer kinds usingdifferent delaying mirror kinds follows.

A. Michelson-class interferometers using realimaging

Superimposing Michelson-class interferometers areshown in Fig. 6A, 6B and 6C. These have delaying mir-rors that use real imaging to form the A.M. This delayingarrangement is also called a relay lens delay. The A.M.is superimposed with the mirror 88 of the other arm bypartial reflection in the beamsplitter 90. In Fig. 6A, thedelaying mirror is a relay lens system comprising a fieldlens 82, a center lens 84, and a spherical end mirror 86.In Fig. 6B the transmissive center lens is replaced by

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Fig. 7A

Original centerof curvature

Re-imagedcenter ofcurvature

100

102

108 106104

Fig. 7B

A.M. plane

120

118Original image ofmirror 112 surface

112A.M. surface

Final image ofmirror 112 surface

116

114

Figures 7A shows calculation of the center of curvature ofthe apparent mirror for a delaying mirror assembly. Figures7B shows calculation of the apparent mirror location for adelaying mirror assembly.

equivalent spherical mirror 85. This minimizes the num-ber of glass elements to reduce dispersion and chromaticaberration. In Fig. 6C the spherical end mirror is substi-tuted by a plane end mirror 90 and a transmissive lens93. This introduces another degree of adjustment whichcan be an advantage. Then the three lenses 91, 92 and93 form a telescope and can be moved as a group 95.

The A.M. is formed by real imaging of the end mirrorsurface (86 or 90). The imaging is mostly accomplishedby the center lens (84, 85 or 92). The primary purposeof the field lens (82, 91) is to make the A.M. appear flat,but it may also contribute to the imaging of the A.M.surface if the field lens is not exactly at the A.M. Usuallythe A.M. is desired to be flat, so that in general the cen-ter of curvature of the end mirror should be imaged toinfinity by the optics of the delaying mirror assembly. Ifthe center lens 84 lies near the center of curvature of theend mirror 86, then most of the responsibility for imagingthe center of curvature lies with the field lens 82. Typ-ically, the center lens is approximately midway betweenend mirror and A.M., and the field lens is near the A.M.However it is optimal if the center lens is not exactly inthe middle, because this reduces focusing adjustability.

Figure 7A and 7B shows how to calculate the A.M. cur-vature and position. The center of curvature 100 (CC) ofthe end mirror 104 is re-imaged by the optics 106 and 108

to form the CC of the A.M. The inset to Fig. 7A showsthat the center lens may slightly shift the initial CC lo-cation 100 to an intermediate point 106. This slight dis-placement must not be ignored in the calculation. Thento make a flat A.M., the focal point of the field lens 108is arranged to coincide with the intermediate point 102so that the final CC is at infinity. To calculate the A.M.location, the image of the end mirror surface 112 is firstre-imaged by the center lens 114 to an intermediate point118. The inset of Fig. 7B shows the field lens 116 re-images point 118 to point 120, which then defines theapparent mirror location. This slight shift from 118 to120 must not be ignored.

The interferometer delay time is given by τ = 2D/c +τglass, where D is the difference in single-trip lengthsof the two arms measured from the beamsplitter whereit intersects the optic axis to the end mirrors 88 and86. The term τglass is the insertion delay contributed byglass components measured down the optic axis, givenby τglass = 2T (n − 1), where T is the glass thicknessand n is the refractive index. This glass term is muchsmaller than the 2D/c term, but can add dispersion dueto the wavelength dependence of n. Thus, the glass termmust be considered when discussing dispersion matchingbetween two interferometers. Otherwise it can usually beignored.

It is preferred that the delaying mirror be as achro-matic as practical. This can be achieved by havingthe lenses and mirrors in the assembly be individuallyachromatic, by using first surface mirrors and standardmulti-element achromatic lenses. In some cases, non-achromatic elements having chromatic effects of oppositepolarity can be combined so that the net chromatismis reduced. For clarity, the multi-element nature of thelenses are not drawn in some of the Figures. Howeverit is implied that achromatic lenses be used where-everpossible, unless explicitly stated that the lens need be asimple lens.

Spherical mirrors have superior achromatism overlenses, and when used at a 1:1 conjugate ratio, can haveless spherical aberration than a transmissive lens for largenumerical apertures. However, the use of a curved mir-ror as the center lens 85 requires an off-axis reflectionwhich introduces undesired astigmatism. (This astigma-tism and other irregularities can be greatly reduced byusing a differential configuration described later.)

Delaying mirrors made from real imaging optics suchFig. 6A, B and C are useful because they provide a prac-tical way of generating long achromatic delays, such asseveral meters in length, having a reasonably large nu-merical aperture, and having an adjustable delay value.The numerical aperture (f/number) is approximately setby that of the center lens and could be f/10, for exam-ple. In contrast, glass etalons longer than 30 centimetersand having a sufficient diameter to achieve a similar nu-merical aperture are impractically expensive, and haveprohibitively large dispersion for broadband use. Fur-thermore, their delay time can’t be adjusted practically.

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The end mirror of either the short or long arms (88or 86) can be tilted so that the A.M. is inclined relativeto the S.M., by a small enough amount that the super-position is not significantly degraded. This generates adelay value τ which is a function of transverse locationacross the beam at the input/output plane. This is calledan inclined delay. When fringes are present in the out-put, this creates an evenly spaced fringe comb of parallelfringes. This is useful for determining fringe phase, be-cause all portions of the sinusoidal fringe shape can beobserved simultaneously, allowing separation from anyconstant component. In contrast, in a non-superimposingMichelson, the fringes are rings due to the dependence ofτ on ray angle. Methods of creating large values of incli-nation without sacrificing A.M. overlap are discussed ina later section.

B. Manifold of element positions for a relay delay

There are a large variety of lens/mirror spacings, choiceof number of elements, selection of focal lengths/mirrorcurvatures which will produce a flat A.M. by real imag-ing. This variety produces freedom to adjust the A.M.-to-end mirror spacing while maintaining a constant cur-vature A.M.. Including more lenses/mirrors yields evenmore degrees of freedom. The freedom to adjust can beused either to modify the delay time τ while maintain-ing the superposition condition, or to adjust the overlapquality while maintaining a constant delay τ .

Two example calculations are given, one for the config-uration of Fig. 6A, and one for Fig. 6C. A calculation forFig. 6B would be similar, with the exception that move-ment of the center mirror changes the total path lengthas well as the spacing between elements. The configura-tion of Fig. 6C has an extra degree of freedom allowedby substituting two elements (lens and plane mirror) forthe single curved end mirror. The focal lengths for theFig. 6C example were chosen to be asymmetrical, that is,the center lens is decidedly more asymmetrically placedbetween the end mirror and A.M. than in the calculationof Fig. 8A.

The calculation for configuration Fig. 6A are shown inFig. 8A and 8B. These are the manifold of possible posi-tions of the optics which produce a flat apparent mirror,calculated for the specific example of field lens 82 focallength = 100 cm, center lens focal length 50 cm, and endmirror radius of curvature 100 cm. In both Fig. 8A andB, the A.M. to end mirror distance is plotted versus cen-ter lens position. However, Fig. 8A the A.M. is fixed, andin Fig. 8B the end mirror is fixed. Figure 8A would bemore useful for when the superimposing overlap must bemaintained while the delay time τ is being changed, suchas in a Fourier Transform spectrometer with a scannabledelay. Figure 8b would be more useful in a double in-terferometer application where the overlap quality of asecond interferometer is being adjusted while trying tomaintain a specific delay, (such as the delay value de-

fined by the first interferometer).Stagnant configurations should be avoided, since they

have no adjustability. The stagnant configurations arewhen the derivative of the A.M.-end mirror distance iszero. These configurations are indicated in Fig. 8A &B by the thick arrows. Generally, they occur when theoptics is in, or near to, a symmetrical configuration. Forthis reason, the magnification performed by the centerlens should usually be arranged to be something otherthan exactly unity.

Figures 9A and 9B show the manifold of configurationsproducing a flat A.M. when the spherical end mirror isreplaced by a plane mirror 132 and a second field lens136. The additional element 136 introduces another de-gree of freedom, so that the complete manifold must berepresented by a 3-dimensional chart showing configura-tion versus two parameters, which could be the centerlens position and one of the field lens positions. For sim-plicity, only a subset of the possible configurations areshown in Fig. 9A and 9B, using 2-dimensional charts.

Since both ends 130, 132 of the cavity are intended tobe flat, the positions of the A.M. and end mirror couldbe interchanged. Secondly, partially reflecting mirrorscould occupy both the A.M. and the end mirror so thata recirculating cavity is formed. This is also true of theconfigurations in Fig. 8A and 8B. Thus this or analogouscalculations could apply to recirculating cavities, whichwill be discussed later.

In Fig. 9A, 9B the focal lengths were chosen to asym-metrically place the center lens between the two fieldlenses, to illustrate that this avoids stagnation points.Field1 136 focal lengths = 30 cm while field2 lens focallength = 90 cm. Center lens 134 focal length = 30 cm.The A.M-end mirror separation with the A.M. positionfixed is plotted versus center lens position. Figure 9Ashows the case where the Field1 lens 130 is fixed at theA.M.

Figure 9B shows the case where the three lenses 130,134, 136 are moved together as a group. This is an easymethod of adjusting the relay delay length while keep-ing the A.M. constant. The three lenses are mountedon a platform 95 (Fig. 6C) which can be moved relativeto the end mirror. The three lenses form a telescope.The three lenses are first spaced properly amongst them-selves so that collimated light incident to the telescopeis outputted from the telescope as collimated. This canbe accomplished by illuminating the interferometer withHeNe light and obtaining a fringe comb of parallel fringesinstead of rings. The parallelism of the fringe comb in-dicates that the A.M. is flat. Once the A.M. is flat, thetelescope 95 can be translated relative to the end mirrorin the fashion indicated by Fig. 9B to adjust the A.M.-end mirror spacing.

Displacement of the telescope 95 only affects the A.M.-end mirror spacing if the telescope comprises lenses withasymmetrical focal length values (field1 = field2 focallength). The greater the asymmetry, the greater thechange of the A.M.-end mirror separation per telescope

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250

200

150

100

50

0

Loca

tion

(cm

)

200180160140120100806040Center lens location (cm)

Apparent Mirror plane

Center lens

End Mirror

Field lens

Configurations producinga flat Apparent Mirror.A.M. fixed

Fig. 8A

Stagnant points

-250

-200

-150

-100

-50

0

-140 -120 -100 -80 -60

Stagnant points

Loca

tion

(cm

)

Center lens location (cm)

Apparent Mirror plane

Field lens

Center lens

End Mirror

Configurations producinga flat Apparent Mirror.

End Mirror fixed

Fig. 8B

Figure 8A shows a manifold of configurations for a flat delaying mirror, when positions are measured from the apparent mirror.Figure 8B shows the same manifold of configurations as Fig. 8A, when positions are measured from the end mirror.

displacement. For example, in the symmetric case wherefield1=field2 focal length, there is zero such adjustability.

C. Applications for an adjustable delaysuperimposing interferometer

An interferometer with an adjustable delay time is use-ful in Fourier Transform (FT) spectroscopy, or as one

interferometer in a pair of matched interferometers forwhite light velocimetry or communications. The FTspectroscopy application requires scanning the delay overa significant range. The matched-pair interferometer ap-plication requires adjusting the second interferometer tomatch the delay of the first, while preserving the super-imposing overlap condition. Previous superimposing in-terferometers (spherical F-P or etalon delayed Michel-son) did not have a practical method to adjust the delay

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250

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150

100

50

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504540353025Center lens location(cm)

Pos

ition

(cm

)

Fig. 9A

Field2

Center

Field1

EndAsymmetric focal lengths.Configurations producing flat A.M.

.Field lens fixed.132

130 Apparent mirror plane

134

136

200

150

100

50

0

403530

Pos

ition

(cm

)

Center lens location(cm)

Fig. 9B

3 Lenses moved as group.

End

Field2

Center

Field1 Apparent mirror plane

Configurations producing flat A.M.

130

134

136

132

Figure 9A shows a manifold of configurations for a flat delaying mirror, for an asymmetrical group of lenses and when one fieldlens is stationary. Figure 9B shows a manifold of configurations for a flat delaying mirror, when 3 lenses are moved as a unit.

over a significant range. Consequently, the interferometercommonly used for FT spectroscopy is the conventionalMichelson, and because this is non-superimposing it hasa limited solid angle over which it can produce a uniformphase fringe. Now the superimposing interferometer us-ing a relay delay can be used for FT spectroscopy. Thisincreases the available solid angle at high spectral reso-lutions (which require long delays), so that wide anglesources such as aurora can be viewed).

It is advantageous to use a superimposing interferom-eter for FT spectroscopy of wide-angle sources becausethen fringe rings are not produced. Instead, a single in-finitely wide fringe can be produced for use with a singlechannel detector to boost signal strength. Or, a uniformfringe comb can be produced for use with a multi-channel

detector. The fringe comb is simpler to analyze than afringe ring.

The delay is scanned by moving the elements of therelay delay in a coordinated fashion set by a scheduleanalogous to Figs. 8A, 9A and 9B, where the A.M. isheld fixed. The elements, individually or in groups whereappropriate, can be mounted on translation stages thatare controlled electronically or mechanically to follow theschedule. Fringes from a reference wavelength, such as aHeNe laser, can be used to monitor the A.M. curvatureand tilt during the scan and correct for deviations bymounting optics on piezoelectric tilt actuators and usinga feedback control loop. The change in τ can also bemonitored by counting passing fringes.

5-14

A.M. surface

BS270 271 272 273 274

275

276

277Intermediate A.M. surface

Fig. 10A

286

287 288

Distributed Mirror

284

281 282 283

A.M.

Fig. 10BFig. 10C

Red Blue

Distributed Mirror

A.M.Etalon

Figure 10A shows a delaying mirror comprising a relay lens delay with added stages to increase its length. Figure 10B showsa delaying mirror implemented by a waveguide and a distributed mirror which compensates for chromatic aberration in thewaveguide. Figure 10C Red and blue rays passing through an achromatic etalon formed from the combination of an etalon anda distributed mirror.

D. Multi-stage relay delay and waveguide

Figure 10A shows a method of increasing the lengthof the relay delay system by adding more stages. Thatis, besides the A.M. 276, one or more additional inter-mediate images 277 of the end mirror 274 are createdthat are intermediate in position between the A.M. 276and the end mirror 274. The A.M. 276 and intermedi-ate A.M. surfaces 277 do not have to be near any lens,as shown, they could lie in between lenses. The detailsof the number of lenses, their spacings and focal lengthsare not important as long as the last apparent mirror 276produced has the same curvature as the short arm mirrorand significant vignetting does not occur along the chainfrom 274 to 270.

The logical extrapolation of the multi-stage long relaychain of Fig. 10A is a waveguide (Fig. 10B), since thisalso forms intermediate A.M. surfaces 281, 282, 283 be-tween the apparent mirror 284 and the distributed endmirror 286. A waveguide is a conduit having a refrac-tive index that varies quadratically cylindrically from thecenter axis. This could also be formed from an infinitenumber of infinitely weak positive lenses. A ray travel-ing through this waveguide makes sinusoidal like pathsthat periodically come to foci (at 281, 282, 283). Thesewaveguides are commercially available in short lengths.In principle, a very long waveguide could be constructed.

These waveguides require a distributed mirror to coun-teract the dispersion in the medium. For example, if asimple plane mirror was placed at the end of the waveg-uide, the apparent mirror would be at different positionsfor different wavelengths. This chromatic dispersion canbe compensated for by using a distributed mirror 286 forthe end mirror, where the effective position of the endmirror for red light 288 is different for blue 287 and ex-

actly compensates the dispersion so that the apparentmirror 284 is at the same location for all wavelengths.

The distributed mirror could be constructed using vol-ume holographic or interference filter technology. Thedistributed mirror would consist essentially of a stackof dielectric mirrors, each reflecting a narrow range ofwavelengths and positioned at the appropriate depth tocompensate for the dispersion of the waveguide, or otherdelay optics. Note that the method of using a distributedmirror could be used in any of the optical designs dis-cussed in this document, not restricted to the waveguidekind, to correct for chromatic dispersion of the A.M..For example, the distributed mirror could be combinedwith an etalon to make an achromatic etalon as shownin Fig. 10C.

E. Delaying mirrors using virtual imaging

Delaying mirrors can be made using virtual imaginginstead of real imaging. Virtual imaging generally occurswhen the spacing of optical elements is closer than theirfocal lengths. One kind of this delay can be called a “2-element” delay to distinguish it from 3-element designs.

An achromatic 2-element delay can be made from thecombination of an achromatic lens and a curved mir-ror. Fig. 11A shows a 2-element delay comprising a pos-itive lens 303 which is close to a convex mirror 302, andFig. 11C comprising a negative lens 305 which is closeto a concave mirror 306. In both kinds, the focal point301 of the transmissive lens 303 or 305 is arranged to su-perimpose the center of curvature (CC) of the sphericalend mirror 302 or 306. This images the CC to infinity, sothat the apparent mirror formed is flat. Other A.M. cur-vatures can be obtained by appropriately adjusting the

5-15

Fig. 11A

Fig. 11B

312 310

A.M.

303

Fig. 11C

301CC

305306

301

302

304

303

Center of curvature(CC)

Figure 11A shows a 2-element delaying mirror using a positivelens, a convex mirror and virtual imaging. Figure 11B showsthat the apparent mirror position for the delaying mirror ofFig. 11A is a virtual image of the end mirror. Figure 11Cshows a 2-element delaying mirror using a negative lens, aconcave mirror, and virtual imaging.

CC

Fig. 12A

314 313

CC

Fig. 12B

316 315

Figure 12A shows a 3-element delaying mirror using virtualimaging. Figure 12B shows another 3-element delaying mirrorusing virtual imaging.

spacings between elements. Figure 11B shows that thelocation 310 is found by virtually imaging the end mirrorsurface 312 by lens 303.

Figures 12A and 12B show 3-element delays describedby Pierre Connes (“Deuxieme Journee D’ Etudes Sur LesInterferences,” Rev. Optique Theorique Instr., vol. 35,p37, 1956). These use a plane mirror and two lenses tocreate an apparent mirror. The combination of lens 314and plane mirror 313 acts as a convex spherical mirror

Fig. 13AA.M.

CC

D<0T

390

392

Fig. 13BA.M.

CC

T

394

Figure 13A is an example of when the apparent mirror is farbehind the end mirror. Figure 13B is an example of when theapparent mirror is very near the end mirror.

analogous to 302, and lens 316 and plane mirror 315 actsas a concave spherical mirror analogous to 306. The ad-vantage of the 2-element configuration over the 3-elementis the smaller amount of unwanted reflections at the lenssurfaces, the smaller thickness of glass the light has totravel through, which reduces unwanted dispersion in thedelay time, and simpler construction and alignment, sinceone less element is required.

Both positive and negative delay times can be createdby the 2-element delay. The net delay time τ = 2D/c +2T (n− 1)/c, which is a sum of a contribution due to thedisplacement D of the A.M. relative to the end mirror,and a contribution from an insertion delay due to thethickness of the glass lens. The former contribution canbe positive or negative. The latter contribution is alwayspositive. The former contribution can be positive whenthe A.M. is in front of the end mirror, as in Fig. 13C.The former contribution can be negative when the A.M.is behind the end mirror, as in Fig. 13A. Figure 13Bshows that a thick lens 394 can produce a positive delayeven though the A.M. is slightly beyond the end mirror.

These delays are optimally achromatic by using lensesand mirrors which are individually achromatic, such asstandard multi-element achromatic lenses and first sur-face mirrors. Alternatively, non-achromatic elementshaving chromatic effects of opposite polarity can be com-bined so that the net chromatism is reduced. This isshown in Fig. 14 which shows the paths 323, 325 of redand blue rays, respectively, and how the chromatic dis-persion of a simple lens 320 can be compensated for bythe dispersion of an etalon 322 of a sufficient thickness.The simple lens will refract the blue more than the redtoward the optic axis. The etalon will refract the blue sothat it has a shallower angle within the etalon. The net

5-16

Fig. 13C

A.M.

CC

T

D>0

392

Figure 13C is an example of when the apparent mirror is infront of the end mirror.

Fig. 14

Red

Blue

Red

Blue

CC

320 322

323

321

Figure 14 shows how the chromatic dispersion of a simple lensand an adjustable etalon in a delaying mirror can cancel.

effect is to have both the red and blue share the samefocal point, which is arranged to coincide the CC of theend mirror 321. The etalon thickness can be adjusted byusing a pair of wedges 322 and 323 having equal anglesthat slide against each other, or by using one or moreetalons in the path and tilting them.

F. Wedge and inclined delay

In many applications it is useful to have a delay valuewhich varies spatially across the apparent mirror, usuallylinearly, so that an interferometer may produce a fringecomb having parallel fringes. This allows correspondencebetween a spatial position in the output image and a spe-cific delay value. This might be called an “inclined” de-lay. A large inclination would be useful for example inFourier Transform (FT) spectroscopy for taking a snap-shot fringe record spanning a large delay range. Thisallows measuring an accurate fringe record for a pulsat-ing source. (Conventional FT spectrometers mechani-cally scan the delay, which takes time and therefore isn’tsuitable for pulsed sources.)

702700

708A.M. #2

701

710

A.M. #1

Fig. 15A

704

706

Mirror ordelaying mirrorassembly 705

T 714

B 712

Fig. 15B

A.M.#2709

713

711

Ray paths

Geometricconstruction

Figure 15A shows the use of a wedge to create a delay whichvaries transversely across the beam. Figure 15B is a detailof wedge showing ray path entering normal to apparent mir-ror, and a geometric construction locating apparent mirrorsurface.

For small degrees of inclination, the incline can beachieved by tilting an interferometer end mirror so thatthe overlap between the A.M. or the end mirror of eacharm varies across the beam. However, for large degrees oftilt this is unsatisfactory because the larger separationsdegrade the fringe visibility, so visible fringes would onlyoccur in some section of the A.M. where the overlap sep-aration passes through zero.

Figure 15A shows a method of maintaining the A.M.superposition in spite of a large mirror tilt. This is amethod of achieving a large delay incline without de-grading superposition quality.

A wedge prism 700 is inserted into the optical path.Since this deviates the output beam path of one arm rel-ative to the others, perfect superposition of output paths(as in Fig. 1B) will not be achieved. However, super-position of images can be achieved (as in Fig. 1C andD) if the wedge is located near the input/output plane.The input/output plane is usually best located near theapparent mirror. Thus the wedge could be located nearA.M. 706, or the end mirror 702, or near the end mirror704 of the delaying mirror assembly 705, since this is im-

5-17

aged to the A.M.. The detector which records the fringesshould be placed at an image of the input/output plane,so that paths from the different interferometer arms con-verge on the same pixels.

Note that in some applications it is desired to havean interferometer having an inclined delay that passesthrough zero delay or its vicinity. To achieve this, anordinary mirror is substituted for the delaying mirror 705.

Figure 15A shows the wedge next to the short armmirror 702. The angles in Fig. 15A may be exaggeratedfor clarity. Figure 15B shows a detail of the action ofthe wedge on rays. The left side of Fig. 15B shows thepath of a ray (bold arrows) incident perpendicular to theA.M. #2 (709). The right side is a geometric constructionfor locating the second apparent mirror A.M. #2 (709),assuming the end mirror 713 is immediately adjacent tothe wedge. This is done by pretending that the surface711 is the first surface of a etalon and that the mirror 713is embedded inside the etalon. For the same reason thatthe etalon 76 in Fig. 4A displaces the apparent position ofthe mirror 72 due to refraction, the refraction at surface711 makes each point on the surface of mirror 713 appearcloser to surface 711 by an amount B that is proportionalto the wedge thickness as measured from surface 711.This creates an apparent mirror A.M. #2 (709) which isinclined to both surface 711 and to the mirror 713. Theapparent displacement is B = T (n − 1)/n at 712, wheren is the refractive index, and T at 714 is the thickness ofthe wedge normal from surface 711. In Fig. 15A, the endmirror 702 and wedge 700 combination is tilted so thatA.M. #2 (708) superimposes the first apparent mirror706 in reflection about the beamsplitting surface 710.

This creates an inclined delay time which varies lin-early transversely along the apparent mirror, while sat-isfying the superimposing condition for monochromaticlight. The inclination magnitude ∆cτ , which is thechange in delay across the beam, is given by ∆cτ =2[T (n − 1)/n + T (n − 1)] is the sum of the componentdue to apparent mirror displacement, and the compo-nent due to the slower speed of light through the wedgethickness. This formula is accurate for small angles ofincidence where sin θ ≈ θ.

Due to wedge glass dispersion, the A.M. overlap is per-fect only for a single wavelength. However, the amountof chromatic mis-overlap is only approximately 2% of thedelay, so it is much smaller than if the mirror were tiltedalone to create the same amount of incline. Since a cer-tain amount of mis-overlap can be tolerated, this methodallows a 50 times greater tilt to be created for the sameloss of fringe visibility.

An example follows: suppose we desire a round tripdelay incline of 1000λ across the A.M. If the wedge re-fractive index is n = 1.5, then ∆cτ = 1000λ ≈ 2T (0.83).Thus we need a thickness T of 600λ ≈ 300 microns forgreen light where λ = 500 nm. Figure 16 shows thatsuch a shallow wedge can be conveniently made by com-bination of two component wedges 410, 412 whose wedgeangles may be slightly different, or whose refractive in-

702

Mirror ordelaying mirror

705

Etalon 718

Wedge/wedge719

Fig. 15C

Figure 15C shows an achromatic inclined delay achieved by awedge combination in one arm and a compensating etalon inanother.

Fig. 15D

blue

redblue

716

717 red

red

blue

Fig. 15E

Figure 15D shows the apparent mirror surfaces for red andblue light when the elements of the wedge combination areconsidered separately. Figure 15E shows the parallelism be-tween red and blue apparent mirrors when the combinationof 15D is considered as a whole.

dices may be slightly different. The net wedge inclineis the sum of component inclines, and can be adjustedby twisting the component wedges relative to each otherabout the common optic axis 414.

Birefringent wedges can be used to create an inclinewhich has different values for different polarizations.This is useful in a polarization resolving retro-reflectinginterferometer, discussed later in Fig. 31, to produce aninclined delay for the detected light, but a uniform delayfor the illuminating light, even though the detected andilluminating light pass through the same interferometerapparatus.

G. Achromatic inclined delays

An achromatic inclined delay can be made use of twoor more wedges in a combination 719 in one arm to-gether, with an ordinary etalon 718 in the opposite arm

5-18

(Fig. 15C). The wedge elements are made of materialshaving differing dispersive powers, such as crown and flintglass, analogous to the method of making an achromaticlens. (Dispersive power is the ratio (nB−nR)/(nB +nR),where nB and nR are the refractive indices for blue andred light. “Flint” glass has a greater dispersive powerthan “crown” glass.) The component wedges are orientedthin side to thick side, as shown in Figure 15D.

Figure 15D shows the wedge combination when theA.M. of each component wedge are considered individ-ually, and Fig. 15E show the net effect when they arecombined. For any dispersive material, because the re-fractive index for blue and red differ, the A.M. formed forred light differs from that formed for blue. The red andblue A.M. for each individual wedge 716, 717 are drawnas dashed lines. They make a slight angle to each other.The goal is to match the red/blue angle in wedge 716 tobe opposite that in wedge 717. When this is achieved,then the net A.M. for red and blue for the combination ofwedges are parallel to each other, as shown in Fig. 15E.The wedge with the more dispersive glass will be thethinner of the wedges.

Since both the red and blue A.M.s are inclined relativeto the thickness variation of the net wedge combination,an inclined time delay is created. An etalon 718 is placed

in the other arm with an appropriate thickness to can-cel the parallel displacement between the blue and redA.M.s.

Specifically, let n1B, n1R, n2B, n2R be the refractiveindices for blue and red light for two wedge materials #1and #2, and T1 and T2 be the thickness of the wedges attheir thick sides, and let the thin side of the wedge havezero thickness, then we require

T1

[n1B − 1

n1B− n1R − 1

n1R

]= T2

[n2B − 1

n2B− n2R − 1

n2R

](5)

in order to make parallel blue and red A.M.s. Then weselect the etalon 718 on the other arm having a thicknessT3 and refractive indices n3B, n3R by requiring

T3

[n1B − 1

n1B− n1R − 1

n1R

]= T2

[n2B − 1

n2B− n2R − 1

n2R

](6)

If α is a parameter describing the fractional positionacross the wedge, then the net delay cτ varies linearlywith α as

cτ = α2T1

[(n1 − 1

n1

)+ (n1 − 1)

]+ (1 − α)2T2

[(n2 − 1

n2

)+ (n2 − 1)

]− DE (7)

where DE is the constant delay contributed by the etalonin the opposite arm and is

DE = 2T3

[(n3 − 1

n3

)+ (n3 − 1)

](8)

and the indices n1, n2, n3 are color dependent refractiveindices for materials #1, #2 and #3.

Realistic wedges have non-zero thickness on the thinside. These can be used in the above equations by con-sidering them to comprise a wedge with a zero thicknessthin side, plus an etalon. These new etalons add an addi-tional constant displacement between red and blue A.M.,so the etalon 781 on the other side must be increased inthickness appropriately to cancel this out.

H. Achromatic Etalon Delays

A conventional etalon delay creates an A.M. that hasdifferent locations for red and blue light due to disper-sion in the refractive index. This prevents application ofthick etalons in broad bandwidth superimposing interfer-ometers. A method of creating an achromatic etalon hasalready been discussed in Fig. 10C using a distributed

Fig. 15F

Etalon #3 "Flint"

Etalon #2 "Crown" 680

682

Figure 15F shows etalons having different dispersive powersplaced in opposite arms to form an achromatic etalon delay.

mirror. An other method is by using two etalons 680,682 placed in opposite arms (as shown in Fig. 15F) andhaving materials of different dispersive powers, such as“crown” and “flint” glass. In analogy with an achromaticwedge discussed above, the thickness of the etalons 680,682 are chosen so that the red-blue A.M. displacementcreated by etalon 680 is canceled by the red-blue A.M.displacement by the other etalon 682. Thus if etalons680 and 682 are identified by subscripts “3” and “2”,then we require that the etalon thickness are related by

5-19

Fig. 17A

D1

D2

∆D

A.M.

A.M.

332

330

335

331

336


DelayingMirror

Assembly

A.M.

A.M.

Fig. 17B

332

331

330

Figure 17A shows a differential superimposing interferometerusing real imaging in the delaying mirrors. Figure 17B showsthat the apparent mirrors of Fig. 17A can be located in thevicinity of the beamsplitter to reduce effect of beamsplittersurface irregularities.

the same formula Eq. 6, and the net delay created willbe described by the formula for cτ stated in Eq. 7, butwith α = 0. The etalon of the less dispersive materialwould be thicker than the etalon of the more dispersivematerial.

A consequence of creating delays using chromaticallydispersive media such as glass is that even though goodoverlap is achieved for all colors (so that fringe visibil-ity is good), the value of delay τ may be a function ofcolor. This is called dispersion in τ . Thus, the term“achromatic” in this document refers primarily to a wave-length independent A.M. overlap, and only secondarilyto a wavelength independent value of τ . Dispersion in τis not a serious problem, because if the fringe is visible,its phase can be adjusted during data analysis after it isrecorded. Secondly, in pairs of matched interferometers,the dispersion in τ can often be matched, so that thefringe phase will not be affected.

Etalon delays are useful because the delay value is in-dependent of transverse position of the etalon, which sim-plifies alignment compared to delaying mirrors that uselenses.

I. Differential interferometer

A method of creating a superimposing interferometerwith a relatively small or zero delay, which can be ad-justable, and which has a wide distortionless image field,is to have a delaying mirror in both arms. This is calledthe differential interferometer. Figure 17A shows a dif-ferential embodiment having delaying mirrors 335 and336 which use real imaging. The apparent mirrors 330and 331 are overlapped in partial reflection of the beam-splitter 332. The two arms may be only slightly differ-ent in length (D1 and D2) and the difference in length∆Dforms the net delay, τ = 2∆D/c. A key advantage isthat since D1 and D2 can be much longer than the dif-ference ∆D, the wavefront curvatures inside the delayingmirrors 335, 336 can be very shallow (nearly flat) whichreduces aberrations in the A.M. they produce. Secondly,these aberrations tend to be similar in shape and mag-nitude, and thereby improve the overlap between A.M.330 and A.M. 331 compared to if one of these A.M. werebeing overlapped with a flat mirror.

This is an advantage, for example, when using an off-axis reflection in the delaying mirror (such as in Fig. 6B)which introduces astigmatism. Since the astigmatism ineach arm will nearly match, then net aberration is re-duced. Similarly, chromatic and spherical aberrationscan be reduced. The result is that the differential con-figuration produces a wider diameter region at the A.M.having good overlap than a non-differential configuration,for the same delay time.

For example, suppose the goal is to generate a 1 cm(single-trip) net delay. If this is done by a single ∼1cm long delaying mirror assembly in one arm overlappedwith a flat mirror in the other, then the aberrations atthe A.M. grow very rapidly with radius away from theoptic axis. (This is because the center lens of a 1 cm realimaging delay would have a small focal length of only 1/4cm.) The “good overlap region” may only be 1 or 2 mil-limeters radius. However, if a differential configuration isused with a 400 cm length relay compared against a 401cm relay, then the wavefront curvature inside each de-laying mirror assembly is long. This creates a very largegood overlap region of many centimeters radius from theoptic axis.

J. Increasing fringe visibility by A.M. placement

In the differential configuration, the apparent mirrors330 and 331 can be located within the beamsplitter asshown in Fig. 17B. In the non-differential configurationthis is not possible, because the A.M. must overlap theshort arm mirror, which is necessarily outside the beam-splitter. Having the A.M. overlap the beamsplitter maxi-mizes the fringe visibility in the presence of irregularitiesin the beamsplitter surface. The explanation is shown byFig. 18A & B.

In a non-superimposing Michelson interferometer

5-20

Fig. 18A

337338

339

Fig. 18B

Delaying Mirror

343344

Delaying M

irror

346

Figure 18A shows how parallel beams in a conventional inter-ferometer are effected by surface irregularities from all of thebeamsplitter area. Figure 18B shows that the beams in thesuperimposing interferometer from a given object pixel maysample only small portions of the beamsplitter.

(Fig. 18A), the beamlet 338 from an object pixel 337 fillsthe entire beamsplitting surface because the rays must beparallel to produce a coherent delay. This is also the caseanytime the A.M. is far from the beamsplitter. The widebeamlet samples the full range of irregularities across thebeamsplitting surface. This reduces fringe visibility.

In contrast, by placing the A.M. at the beamsplitterand by making the A.M. the input/output plane, thenthe beamlet of rays 346 leaving an object pixel 343 passesthrough a minimal area of the beamsplitting surface, sothat over this small region the beamsplitting surface isuniform so the fringe visibility is high. Thus, any irreg-ularities in the beamsplitter cause a irregularities in thephase of the fringe and not its visibility. This is better,because the phase irregularities are constant and can becorrected numerically at a later time.

K. Recirculating interferometer

The Michelson interferometer is a two-path interferom-eter which generates one echo in addition to the originalpulse, for a given applied pulse. Another kind of interfer-ometer, called the recirculating interferometer, generatesan infinite progression of gradually decreasing echos all

Fig. 19A

250254 252 251 253255

256257

Fig. 19B260 261

Figure 19A shows a simple superimposing recirculating inter-ferometer. Figure 19B shows use of a center lens combinationto increase adjustability.

Fig. 19C

266 268

264260 262

Field1 Center Field2

End2End1

Fig. 19D

250800 802

256

255

804

257

Figure 19C shows a recirculating interferometer with planeend mirrors and lenses substituted for curved mirrors. Figure19D shows coupling light in and out of a recirculating interfer-ometer using a beamsplitting surface not at the end mirrors.

separated by the same delay τ , which corresponds to theround trip time. A conventional Fabry-Perot is an exam-ple of a recirculating interferometer, however it is non-superimposing. The spherical Fabry-Perot of Fig. 3A isa superimposing, but its delay time is not adjustable be-cause the mirror center of curvatures are fixed and mustoverlap. Secondly, the beam can’t pass exactly downthe optic axis 31 because it will hit the edge betweenthe fully reflective and partially reflective regions 33, 32.This is unfortunate because the minimum aberrations ofthe curved mirrors are achieved for rays going down theoptic axis.

New superimposing versions of a recirculating interfer-ometer are shown in figures 19A, B C and D. Some ofthese versions allow adjustability for the mirror spacing,and hence delay, while maintaining the superimposingcondition. In Fig. 19A, a positive achromatic lens 250,

5-21

called the center lens, lies in between two curved partiallyreflecting mirrors 251 and 252 and images the surface ofmirror 252 to the other mirror 251, and produces positiveunity magnification per round trip so that a ray returnsto its same place on the mirror surface after one circuit.In contrast, the roundtrip magnification of the sphericalFabry-Perot in Fig.3A is not positive unity, but negativeunity, since the ray does not return to the same place onan end mirror after one roundtrip between the two endmirrors. It requires two roundtrips to return to the sameplace.

The center lens 250 can be replaced by a combinationof lens elements to increase the adjustability of configu-ration. In Fig. 19B, two lens elements 260 and 261 actas the center lens.

The curved mirrors 252 and 251 of Fig. 19A & B canbe replaced by combinations of a flat mirror and lens,such as 266 & 260, 268 & 264 in Fig. 19C. In general forFig. 19A, B & C, the center of curvatures (CC) of thetwo end mirrors must overlap after accounting for anyimaging effect by the internal lenses. Thus, in Fig. 19C,left infinity must be imaged by the 3 lenses 260, 262, 264to right infinity. It optimal to have the center lens 250 orlens combination 260 and 261 asymmetrically placed be-tween the end mirrors 251 and 252 to create adjustabilityfor imaging the surface of 251 to surface 252.

At least one partially reflecting surface must be pro-vided so that light can be coupled into and out of thecavity. This can be accomplished by making the mir-rors 252 and 251 partially transmitting/reflecting. Thesetwo mirrors do not have to have the same amount of re-flectivity/transmissivity. Alternatively, an explicit beam-splitting surface such as the beamsplitting cube 804 inFig. 19D can be placed anywhere in the recirculatingpath. This allows the end mirrors to be totally reflect-ing, which is very practical. For an input beam 257, twooutput beams 255 and 256 are created.

The recirculating interferometer has spectral proper-ties that are analogous to the conventional Fabry-Perotused directly on axis. That is, whereas the Fabry-Perothas a circular central fringe which subtends a small coneangle about the optic axis, the superimposing interfer-ometers can have an extremely wide central fringe thatfills the entire field of view. This is a practical advantage.

The spectral character of a Fabry-Perot class of inter-ferometer consists of a comb filter with peaks that canbe very narrow compared to the spacing between peaks.Smaller loss of intensity per round trip produces nar-rower peaks (ie., smaller transmittance of the end mirrorsor larger transmittance of the beamsplitting cube 804).Numerous textbooks describe the spectral behavior ofFabry-Perot class interferometers versus the reflectivityand loss in the cavity. For example, on pages 307-309 inthe book by Eugene Hecht and Alfred Zajac, “Optics”,Addison-Wesley, Reading, Massachusetts 1976, Libraryof Congress No. 79-184159. Thus the reflectivities of theelements in a recirculating interferometer are chosen bydeciding how narrow of a spectral peak is desired (related

to a quantity called the finesse in textbooks).In Fig. 19A, the lenses 254 and 253 outside the re-

circulating cavity cause the wavefront curvature of lightentering and leaving the cavity to match that of light in-side the cavity. This allows a collimated incident beam at257 to pass through the interferometer and be outputtedat 256 as collimated. The focal point of lens 254 wouldbe superimposed with the center of curvature of curvedmirror 252, for example, to achieve this. Similarly forlens 253 and the CC of mirror 251. This is not necessaryfor when the end mirrors are flat, as in Fig. 19C.

A convenient method of designing the optics of a re-circulating cavity is to consider a group of elements onone side of the cavity to form a delaying mirror whichproduces some apparent mirror surface, which may becurved or flat. Then the remaining optic or optics mustsuperimpose with this A.M and face it so that the light re-circulates. For example, the surface and center of curva-ture of mirror 252 must match the surface and CC of theA.M. computed for the combination lens 250 and mirror251 in Fig. 19A. Secondly, the roundtrip magnification ofan image must be positive unity, and not negative unity.Otherwise, there will be upside down images combinedwith the rightside up images in the output. When theseconditions are met, superposition of paths is achieved.

Note that the combination of lenses are internal to thecavity in Fig. 19C should form a telescope that passescollimated light as collimated. By making the lens con-figuration asymmetric, the movement of the telescope al-lows the end mirror 266 to end mirror 268 spacing to beadjusted while maintaining the superimposing condition.Thus, the chart of Fig. 9B applies to the recirculatinginterferometer of Fig. 19C by placing a flat end mirrorafter the field lens 130.

L. Interferometers made without delaying mirrors

Superimposing interferometers can be made withoutdelaying mirrors by explicitly routing the beams usingone or more beamsplitters into the topological configu-ration desired, such as the 2-path topology shown as in-terferometer 3 of Fig. 5A. The advantage is that this canavoid having the beam come back on itself, so that allof the interferometer output beams are easily accessible.In contrast, for the Michelson configuration (Fig. 4A)where the incident beam 77 enters normal to the mirror72, there are two complementary outputs at 73 and 75.However, the output 75 comes back in the same directionas the incident beam 77, making it difficult to access.(One solution is to angle the incident beam off-normal.This is more easily done with a superimposing interfer-ometer than non-superimposing because of the lack ofstrong angle dependence to the delay.)

A large variety of configurations are possible. Figure20A shows one embodiment for a 2-path interferometer.Beamsplitter 836 splits the incident beam into two paths,and beamsplitter 838 recombines them to form two com-

5-22

BS836

Fig. 20A 824

826

828832

834

820830

822

BS838

L1L1

L1L2 L2

841

840

820 822 824 826 828830

L1 L1L1

L2L2

832

834

Long Arm

Short Arm

Fig. 20B

Mag=-1 Mag=-1

Figure 20A shows a superimposing 2-path interferometermade without a delaying mirror by explicitly routing thebeams using relaying optics. Figure 20B shows the relay sys-tems for the long and short arms of Fig. 20A laid out sepa-rately in a line.

plimentary outputs (840, 841). The difference in pathlengths yields the delay time. In order to superimposeoutput paths for a given input ray, a relay lens systemis used generally in one or both arms. In the case ofFig. 20A, it is only used in the longer arm. Any relaylens systems that are used must superimpose the inputplanes for each arm, and superimpose the output planesfor each arm, and match magnifications. Optimally, inorder to superimpose paths it should further match wave-front curvatures. Otherwise, it will only superimpose im-ages, which is useful but less desirable. In Fig. 20A, theinput planes are 834 and 830, superimposed by the beam-splitter 836, and the output planes are 834 and 832, su-perimposed by beamsplitter 834. Note that 834 is bothan input and output plane.

For each arm, the input plane (830 or 834) is imagedby optics to the output plane (832 or 834). This is mosteasily seen by “unwinding” the two optical paths anddisplaying them separately in a line, in Fig. 20B. Since

Fig. 21A

L2

L2

L1

L1

L1

L1

L1

L1

850 852L2

852

Mag=-1 Mag=-1 Mag=-1

L1

L1L2

L2

Long Arm

Short Arm

Fig. 21B

Mag=-1

L1 L1L1 L1 L2

L2

850 852

850 852

Figure 21A shows another superimposing 2-path interferom-eter having a different overall polarity of magnification thanin Fig 20A. Figure 21B shows the relay systems for the longand short arms of Fig. 21A laid out separately in a line.

the input and output planes are flat, we desire that apoint at left infinity be imaged by the relay optics toinfinity on the right. (This is a way of matching wavefrontcurvatures between the two arms.)

The focal lengths and positions of the lenses compris-ing the relay systems can be anything that satisfies theabove conditions. The embodiment shown in Fig. 20Aand B was chosen because it is simple to understand.The lenses 820 and 828 at the ends of the system have afocal length L2 twice the focal length L1 of the internallenses 822, 824, 826. Each internal lens images the cen-ter of the lens to its left to the center of the lens to itsright. Each end lens 820 and 828 focuses collimated lightto/from the center of the lens neighboring it. This par-ticular embodiment of the relay chain can be organizedinto two “stages”, each which produces a magnificationof negative unity. The total magnification of the systemis therefore (−1)(−1) = 1 is positive unity. This matchesthe magnification of the short arm, which is obviouslyunity because there is no relay system in the short arm.

Figure 21A shows a similar 2-path superimposing in-terferometer, that differs from that in Fig. 20A by theinclusion of one more “stage” to the relay system of each

5-23

BS

L1

L2

L1

L1

L1

BS L2

Fig. 22

In Out

L2

L1

Auxilary Out

882

883 881

880 886884

Figure 22 shows a superimposing recirculating interferome-ter made without a delaying mirror by explicitly routing thebeams into a circulating path using relaying optics.

arm. That is, now the short arm has one stage and thelong arm three stages. Thus the magnification betweenthe input 850 and output 852 planes is negative unity forboth paths. Figure 21B shows the relay systems of eacharm of Fig. 21A when they are laid out in a line.

Figure 22 shows a recirculating superimposing inter-ferometer made without a delaying mirror by explicitlyrouting the beams using relaying optics. Internal to cav-ity, the four lenses (880, 881, 882, 883) are chosen andpositioned so that the magnification per round trip is ex-actly positive unity and the wavefront curvature is pre-served per round trip. This is analogous to taking therelay system of Fig. 20B and making it into a circle bybutting up the two end lenses 820 and 828 together. Inthis embodiment the four internal lenses have the samefocal power L1. The external lenses 884 and 886 helpcouple collimated external beams into/out of the cavityand have a focal length L2 twice as long as L1.

M. Zero delay applications of superimposinginterferometers

Although we have been primarily discussing interfer-ometers having non-zero delays, interferometers havingzero or near zero delay are also useful for position orthickness measuring applications. A superimposing in-terferometer is useful in these applications, particularlywhen a mirror needs to be placed internal or at the sur-face of an object when this is physically impossible ordifficult with a conventional interferometer. Figure 23Ashows an application where a mirror surface would ide-ally be placed internal to an object, in this case a fluidcell 902 which contains a transparent sample 900 whose

M2

Fluid cell

BS

M1

Pnt. B

Pnt. A

SampleFig. 23A 900902

904

906

908

910

912

901

M2 A.M. 920


Fig. 23B

Delaying MirrorAssembly

900 Sample

A.M. 922

926

924

BS912

901

Figure 23A shows how rays do not pass through the same spotin a irregular sample if the mirror of a conventional zero-delayMichelson is not close to the sample surface. Figure 23B showshow a delaying mirror can be used in a zero-delay Michelson tomake rays pass through the same spot on an irregular sample.

thickness profile is to be measured interferometrically viatransmission through the whole cell. The illuminatinglight incident along ray 901 may be chosen to have alow coherence length so that the zeroth fringe can be un-ambiguously determined. In this case, it is necessary tomatch the lengths of the two arms, between the beam-splitter 912 and mirror 906, and between beamsplitter912 and mirror 904. If the sample 900 has an irregularsurface profile it will deflect the rays through refraction.If the mirror 904 is not close to the surface of the sample,then a ray passing through point A (908) will not passthrough the same point A after reflecting from the mir-ror 904. This can blur the measurement, since the lightessentially samples two different places on the sample,points A (908) and point B (910). Ideally, the mirror M1

(904) should be as close to the sample’s surface as possi-ble, but this may not be practical because of the windowsof the fluid cell.

A solution to this problem is to use a superimposinginterferometer, such as shown in Fig. 23B. The use of atleast one delaying mirrors allow the apparent mirror 920of the delaying mirror assembly 924 to be placed internalto the fluid cell next to the irregular sample surface 900,because it is not a physically real surface. This way, the

5-24

Delay

Two-pathInterferometer

Impulse response

t

Fig. 24A

Fig. 24B

Fig. 24C

Delay

d

c

N-tupletInterferometer

1

Delay etc.

Delay

Figure 24A is an electrical equivalent of a generic two-pathinterferometer. Figure 24B is an impulse response for a two-path interferometer. Figure 24C is an electrical equivalent ofa N-path interferometer.

light reflecting from the A.M. 920 passes close to the sameplace on the sample surface that it passed incident towardthe A.M. The A.M. should be in front of the elementsthat comprise the delaying mirror assembly. This can bedone with a real imaging system by choosing the focallengths and positions of the elements appropriately. Forexample, in the chart of Fig. 8A at a center lens positionof 160 cm, the apparent mirror is in front of the field lens.

The zero or near-zero delay is achieved by using an-other delaying mirror 926 in the other arm having amatching delay, as in the differential interferometer con-figuration. If long coherence length illumination is used,then the second delaying mirror 926 is not required. In-stead, an actual mirror could be placed at the positionM2 (922). In all cases, it is optimal to superimpose thetwo mirror surfaces 922 and 920 in partial reflection ofthe beamsplitter 912.

N. Electrical circuit equivalent

Figure 24A shows the “electrical circuit” representa-tion of a two-path superimposing interferometer, mean-ing that only the temporal behavior is represented andnot the ray paths. Figure 24B shows the impulse response

Delay

b |b|<1

RecirculatingInterferometer

Fig. 24E

375

t

Fig. 24F

t

Fig. 24D

Figure 24D is an example impulse response for a 3-path in-terferometer. Figure 24E is an electrical equivalent of a re-circulating interferometer. Figure 24F is an example impulseresponse for a recirculating interferometer.

of the optical field. Generally, the spectral behavior ofan interferometer is obtained by Fourier transforming itsimpulse response. Since the superimposing interferom-eter can generate the same impulse response for all theincident rays, the spectral properties can be uniform forthe whole beam. In contrast, the impulse response of anon-superimposing interferometer is the same as a super-imposing kind only for the rays traveling exactly downthe optic axis of the non-superimposing interferometer.That is, the superimposing interferometer has a fringewhich could be infinitely wide and corresponds to the fi-nite diameter center fringe (in a system of fringe rings)of a non-superimposing interferometer. The advantage ofthe superimposing interferometer is the ability to makethe fringe infinitely wide if desired, or into a uniformlyparallel comb of fringes, and not be limited to a systemof fringe rings with a periodicity set by the delay value.

For Fig. 24A, a single applied pulse generates two out-put pulses separated by a delay τ . Figure 24C illustratesan interferometer with more than 2 paths, which could becalled a N-tuplet interferometer with N representing thenumber of paths. For each path beyond the first, a delayis added in parallel to the circuit. The impulse responsefor Fig. 24C has one spike for each of N paths. Figure24D shows the case when N is three. In some applica-tions the output spikes could be separated by different

5-25

delay intervals.A Michelson interferometer usually has just two arms

and so is described by Fig. 24A. If an additionalbeamsplitter is now added to one of the arms (suchas Fig. 26A), splitting it, then we essentially have 3arms. This can be a 3-path interferometer described byFig. 24C. Additional beamsplitters can be further addedto any of these arms, creating more arms, and so on.The apparent mirrors or end mirrors associated with eacharm could be made to superimpose. This is one methodof creating an N-path superimposing interferometer withan arbitrary N. Another method is to use beamsplittersto explicitly route the beams along an N-path topologygiven by Fig. 24C. Relay lens systems can be used to su-perimpose the input/output planes of each path, as inFig. 19A, so that the interferometer is superimposing.Any arbitrary number of paths N can be made. Combi-nations of the two methods are possible.

Figure 24E shows the electrical equivalent to a recircu-lating interferometer. The factor b is a gain of an ampli-fier 375 which represents the loss per round trip incurredat the partial reflection of the mirrors or beamsplitters.The factor b has a magnitude less than unity, to repre-sent loss. The impulse response (Fig. 24F) is an infinitegeometric series of decreasing amplitude spikes, havingan interval given by a round trip time τ .

O. Two-delay interferometer

A superimposing interferometer can be created havingmore than 2 arms by inserting additional beamsplittersinto a Michelson, by using actual mirrors or delaying mir-rors in each arm, and by superimposing all the apparentmirrors and end mirrors using the beamsplitters. Fig-ure 26A shows an embodiment which has a secondarybeamsplitter in one arm, which splits the arm into twosubparts, each which has a delaying mirror (432, 434).Figure 25 is a legend of some symbols used. The ap-parent mirrors 433, 435 of each delaying mirror assem-bly superimpose the short arm end mirror 436 in partialreflection of the main beamsplitter 438. This creates asuperimposing interferometer having two delay values, τ1

and τ2, which can be different.The fringe pattern produced from a two delay inter-

ferometer is a combination of the separate fringe pat-terns from single delay interferometers having delay τ1,τ2 and |τ1−τ2|. This can be useful in velocity interferom-etry to reduce the ambiguity of a velocity determination,even when using one color. In other words, the veloc-ity per fringe proportionality (λ/2τ) is set by both thecolor and the interferometer delay. Having simultane-ous measurements using different proportionalities canreduce the ambiguity of the velocity determination, andthis can be achieved either by using simultaneously dif-ferent colors or delays. It is possible, particularly whenan inclined delay and short coherence light is used in aninterferometer pair, to arrange for the place where fringes

are visible for one delay to be where fringes are washedout for the other delays, so there is no confusion betweenoverlapping fringe patterns.

Figure 26B is the configuration of Fig. 26A where thesecondary beamsplitter 430 has been replaced by a polar-izing beamsplitter 440. When the input beam has inten-sity in both horizontal and vertical polarizations, such asif it is polarized at 45, then two echos 442 and 444 areoutputted having orthogonal polarizations. The unde-layed output pulse 446 has its original polarization. Thisis a method of distinguishing the two different delays ofa two-delay interferometer so that the associated fringepatterns don’t confuse each other. The fringe patternsshare the same output path but are distinguishable bypolarizers placed before the detectors. This could be use-ful in a Fourier Transform spectroscopic application, bymeasuring two delays at once to help distinguish outputfluctuations due to common mode intensity fluctuationsfrom the desired fluctuations due to spectral character.The end mirrors 441, 443 can be tilted independently toproduce a fringe comb with a different spacing for differ-ent polarization.

Figure 26C shows another method of producing a po-larization dependent delay value, where τ1, τ2 differ onlyslightly. The secondary polarizing beamsplitter 450 isplaced in the arm without the delaying mirror. Thissplits the short arm into two paths ending in two endmirrors 452, 454. These are nearly superimposed withthe apparent mirror 456 of the delaying mirror assem-bly. Only small differences in the overlap between mirrors452 and 454 can be tolerated, so only small differencesbetween τ1 and τ2 are possible. Larger differences or in-clinations can be achieved using etalons or wedges nearthe end mirrors 452, 454.

P. Delaying a beam using a delaying mirror

Delaying mirrors have been discussed integral withtheir use in interferometers. However, they can be dis-cussed as an independent element used to delay a beam.Other users may then use this element inside interferom-eters of their own design. For example, in stellar interfer-ometers, the path lengths of the beams from satellite tele-scopes to a central station must be equalized coherently,before they can interfere. This is presently done with-out a delaying mirror by the design shown in Fig. 27A.These delays require parallel light to produce a coher-ent delay between input and output planes 500 and 501.Light from a telescope viewing an extended source maynot be sufficiently parallel for the delay to be coherent.Delaying mirrors are useful because they can coherentlydelay any beam including an uncollimated beam, and thisdelay value may be adjustable.

An important practical concern is how to separate theinput beam from the output beam reflected off the delay-ing mirror. Figures 26B, C and D show different meth-ods. In general, the input and output planes are opti-

5-26

Fig. 25

Legend of symbolsDelaying Mirr.

Delaying mirrror assembly with apparentmirror at left end and end mirror at right end

Normal (nonpolarizing) beamsplitter

Polarizing beamsplitter

Polarizing beamsplitter twisted 45° to planeof drawing

Retarder, usually quarter wave

Mirror

Horizontal polarization

Vertical polarization

Ray direction

Camera or intensity detector

Fig. 26A

Delaying MirrorBS BS

1

2

438 430433

432439

437

434

435

436 Delaying Mirror

Figure 25 is legend of symbols. Figure 26A shows a superimposing interferometer that imprints two echos having a differentdelay times.

mally placed at the A.M. of the delaying mirror. Figure27B shows the input 514 and output beams 515 distin-guished by angle of incidence into the apparent mirror516 at the beginning of the delaying mirror 513. Themaximum practical angle of separation is determined bythe details of the optics internal to the delaying mirror513, such as the numerical aperture of the center lens,if real imaging is used. Figure 27C uses non-polarizingbeamsplitter 503 to separate a portion of the outputbeam 510 from the input path 504. The arrangementof Fig. 27D avoids the backwards output 504 by usinga polarizing beamsplitter 506 (PBS) and a quarter waveretarder 507 with its axis rotated 45 to the vertical axisdefined by the PBS. The light makes two passes throughthe retarder. This rotates the polarization by 90, so thatthe path of the outgoing light is separated from incomingpath by the PBS. The retarder 507 can be replaced byany optical element that rotates by 90 the polarizationof light making a double pass, such as a prism assembly.

For delaying mirrors using real imaging, the delay timecan be changed while holding the apparent mirror it cre-ates fixed by moving the optics internal to the delayingmirror in a coordinated fashion analogous to the chartsin Fig. 8 and 9. By keeping the apparent mirror fixed,

the alignment of the external system using the delay isnot disturbed while the delay value changes.

Q. Superimposing interferometer in series withspectrometer

It is useful to combine a superimposing interferome-ter in series with a chromatically dispersive spectrome-ter (such as prism or grating). This can enhance fringevisibility in spectroscopic and velocimetry applicationsbecause it prevents crosstalk between fringes of differentwavelengths having different phases. The interferome-ter 726 can precede (Fig. 28A) or follow (Fig. 28C) thespectrometer 727. The advantage of using a superimpos-ing interferometer instead of a non-superimposing inter-ferometer, when combined with the spectrometer, is theability to imprint a constant delay or a uniformly inclineddelay with adjustable inclination across the spectrum.

Figure 28A shows the interferometer 726 preceding thespectrometer 727. Since the spectrometer has a slit-likeentrance pupil 723, the light passing through the interfer-ometer should be line-like in cross-section. If the sourceis an optical fiber 720, the light is formed into a line721 by cylindrical optics 722, or by use of a fiber bundle

5-27

Delaying MirrorBS

Delaying Mirror

PBS

For vert. pol.

For horiz. pol.

Fig. 26B

V

H

V

H

446442

444

440 441

443

PBS

BSV

H

V

H

Delaying Mirror

Fig. 26C

456

452

454

450

Figure 26B shows use of polarizing beamsplitter to make a polarization dependent delay. Figure 26C shows a configurationproducing polarization dependent delays that are only slightly different.

500

501

Fig. 27A

Delaying M

irror

Fig. 27B

513

514515

516

Fig. 27C 508

Delaying Mirror

505

BS

504

503

510

/4 retarder 507

Fig. 27D

Delaying Mirror

PBS506

Figure 27A shows a conventional adjustable length delay line.Figure 27B shows an adjustable length delay using reflectionfrom the apparent mirror of a delaying mirror. Figure 27Cshows an adjustable delay line implemented by a delayingmirror and a normal beamsplitter. Figure 27D shows a delayline with a polarizing beamsplitter and quarter wave retarderto avoid backward going beam.

where individual fibers are rearranged into a line. Thisline-like beam 721 is sent into the input plane 724 of theinterferometer, leaves the interferometer at the outputplane 728 and enters the spectrometer system slit 721

as a line-like beam having fringes. The spectrometer isrepresented by the prism symbol 732. The actual de-tails internal to the interferometer and spectrometer areomitted. The spectrometer disperses the incident lightperpendicular to the direction of the slit to form a rect-angular spectrum 726. This spectrum could have fringes,as suggested by Fig. 28B.

The interferometer can imprint either a constant delayor a delay that varies rapidly across the slit-like length, sothat periodicity of the fringes can be arbitrarily adjusted,and could be infinitely wide. In spectroscopy of sourcesthat have a non-smooth spectrum, such as the absorptionlines in sunlight or starlight, the fringes may vary in phaseand amplitude from wavelength channel to channel. Invelocimeter applications, where light from the detectinginterferometer is dispersed by a spectrometer, then thefringes could form a systematically varying pattern versuswavelength and delay (position along the slit).

Figure 28C shows the spectrometer 725 preceding theinterferometer 729. In this case the spectrometer presentsa rectangular spectrum as an input 731 to the interfer-ometer. The interferometer passes this spectrum to itsoutput image plane 733 while imprinting a fringe patternon the spectrum. The orientation of the fringes could bein any direction relative to the wavelength axis, depend-ing on which direction the inclined delay is made.

5-28

Fringes along slitdirection

Fig. 28A

Super. interf.

Input plane724 728

720

722

Output plane

Prism or grating

Spectrometer

Spectrumwith fringes

Line formingoptics

726

727

726

721723

Fig. 28C

Super. interf.

Input plane

720

722

Prism or grating

Spectrometer

Output plane

Spectrumwithoutfringes

Line formingoptics

Spectrumwithfringes

725

729

731 733

Figure 28A shows a spectrometer following a superimposing interferometer chromatically dispersing the fringes. Figure 28Cshows a superimposing interferometer following a spectrometer imprinting fringes in any direction relative to the wavelengthaxis.

Spectrum withfringes

Fig. 28BFringes

736(Delay)

Position along line source

Wavelength

Figure 28B is suggested appearance of a spectrum havingfringes perpendicular to the wavelength axis.

R. Double superimposing interferometer

A system of two superimposing interferometers in se-ries having matched delays and dispersion characteristicsis useful for Doppler velocity measurement and opticalcommunications. Figure 5A is a topological schematic ofa double interferometer for a velocimeter application con-sisting of an illuminating interferometer 3 having delayτ1 and detecting interferometer 7 having delay τ2, with atarget 5 interposed between the interferometers. Whenthe delays match τ1 ≈ τ2 within the coherence lengthof the source 1, partial fringes are produced in the de-tecting interferometer outputs 8, 9. The target displace-ment during the interval τ1, which is an average velocity,is proportional to the fringe phase shift by multiplyingby λ/2τ , for the typical case when the target-reflectedlight and illuminating light are anti-parallel. A changein refractive index along the optical path between theapparatus and the target will also produce a fringe shiftanalogous to target motion.

5-29

1. Optical communication with a double superimposinginterferometer

When the target 5 is replaced by any arbitrary opti-cal path which could be an optical fiber, an open beam,or a beam reflected off obstacles, communication can beachieved between the illuminating and detecting sides.The fringe phase in outputs 8, 9 can be modulated byvarying τ1 slightly about its nominal value τ1 ≈ τ2. Thusa message can be communicated, if the illuminating 3 anddetecting 7 interferometers are matched in delay within acoherence length of the illumination, and matched in dis-persion. (Delay dispersion is the variation of τ with wave-length.) If delay and dispersion are not matched, thensignificantly visible fringes are not produced. This hasbeen discussed in the prior art using non-superimposinginterferometers and called “coherence multiplexing”.

Figure 5B shows an embodiment for optical communi-cation. The use of superimposing interferometers givesthe apparatus the useful ability to use any commonlyavailable light source 930, such as a candle, an incan-descent lamp or sunlight, and the ability to use a widediameter optical fiber 944 or any imaging or non-imagingpath, which can include reflection off buildings or otherobstacles. The delay and dispersion of the illuminating932 and detecting interferometer 934 are arranged to bematched. Optimally, when a non-imaging path 944 isused, there should be no inclination in the illuminatinginterferometer delay, so that all transmitted rays havethe same imprinted delay. This way they can scrambletogether without blurring the imprinted delay.

The signal is sent by modulating the delay of the illu-minating interferometer by a slight amount such as λ/2so that a fringe shift is seen in the detected light of thedetecting interferometer. Analog or digital signals couldbe sent depending on whether the fringe is smaller thanor similar to λ/2; analog modulation would be easier withsmall phase changes. The delay could be modulated by apiezoelectric transducer 938. Alternatively, a mechanicalmeans could be provided so that the pressure of a fin-ger of the operator moves the mirror by a small amountcomparable to λ/4.

By using sunlight, this is a means of communicatingin the outdoors in an encrypted form using little or noelectrical power, which could be useful for military orsurreptitious applications. Only the intended recipientof the message will see fringes in the transmitted beam946. Other viewers lacking the interferometer or lackingknowledge of the proper delay and dispersion will notdetect a significantly visible fringe. Since the phase ofthe optical field is not possible to be recorded by conven-tional detectors (which only measure intensity) withoutthe proper interferometer, it is not possible to record thecoherence properties of the signal and then later analyzeit for an encoded message. The signal must be detectedin real time using an interferometer of the matching delayand dispersion.

Analogous to channels on a radio, if the receiving party

does not use an interferometer with the matching delayand dispersive qualities, no fringes are seen and the lightfrom the transmitter 932 appears innocuous. A varietyof superimposing designs could be used for the interfer-ometers 932 and 934. Figure 5B shows use of achromaticetalons, as discussed in Fig. 15F, along with adjustablethickness etalons 940 and 942 to help match dispersion.Because the method of detecting the communication in-volves the coherence properties of the light, and not theintensity, it is possible to detect the communication evenin the presence of extraneous light such as daylight. Thiscould make the technique useful for surreptitious com-munication by shining the encoded beam on the side of abuilding where it would be hidden by virtue of combin-ing inconspicuously with other natural light. The inter-ferometers could be fashioned into a small portable casesimilar in size to an ordinary pair of binoculars. Theinterferometer pair would be matched initially before be-ing separated and used “in the field”. Since there is alarge number of possible delays and dispersive qualitiesavailable to choose from, it would be unlikely that an un-intended recipient could search for the delay and disper-sion values during the short time that a message wouldbe sent, thus providing the security.

2. Matching delay and dispersion in a doublesuperimposing interferometer

The advantage of using superimposing interferometersfor a double interferometer systems is that this allows thesource 1 to be broadband and extended, such as an in-candescent or flash lamp. It allows the use of wide diam-eter optical fibers, other non-imaging conduits, or poorlyimaging (out of focus or blurry) systems to conduct lightbetween the target 5 to/from the detecting/illuminatinginterferometers. This can be a great practical advantagebecause the sources can then be inexpensive, lightweight,rugged, compact and consume much less electrical powerthan many lasers. Secondly, it allows a large depth offield (range of target motion) because strictly parallellight does not have to be maintained through the detect-ing interferometer.

Since the coherence length of white light is about 1.5micron (a micron is 1/1000th of a millimeter), it can betedious to achieve the matching condition τ1 ≈ τ2 andthe dispersion matching condition while simultaneouslyoptimizing the superimposing overlap in both interferom-eters.

Figure 29A shows a double superimposing interferom-eter embodiment which can achieve this matching, usingtwo superimposing interferometers of the type shown inFig. 6B. The embodiment includes a target 228. Othertargets more remote from the apparatus could be mea-sured by relaying their image through a telescope to theplane 228. The apparatus of Fig. 29A can be used forother double interferometer applications such as commu-nication by introducing a means for modulating the illu-

5-30

PZT

Received signal

Optical Fiber orother path

Signal to send

Light detector

Amplifier

IlluminatingInterferometer

Detecting Interferometer

Lamp orsunlight

FIG. 5B

930

938

932

934

936

940

942

944

946

Figure 5B is a double superimposing interferometer system capable of communication.

Illuminating

Detecting

Target

Lamp

Camera

1:2 gearbox

Translator Translator

202

220

204206208

210

212

216

215

214222 230

218

228

226

224

215

207

205

221

203

219

211

213

217

Fig. 29A

201

A.M.200

Figure 29A shows a double superimposing interferometer with means for matching delay, dispersion while optimizing apparentmirror overlap.

minating interferometer delay and substituting an opticalconduit such as a fiber for the target 5.

The apparatus of Fig. 29A consists of two nearly iden-tical halves; each is a superimposing interferometer sys-tem enclosed in dashed boxes and labeled “illuminating”

and “detecting.” The optics in the two halves are not re-quired to be identical, but this makes it easier to matchdispersive characteristics. Some parts that are not iden-tical between the two halves include the lamp 202, whichis replaced by a camera 203, and the adjustable thickness

5-31

202

204206

210

212

214

215

216

200A.M.

230

222

208

Fig. 29B

230

200A.M.

216

215

222

224

226

228

Fig. 29C

Figure 29B details the subsystem in Figure 29A conveyinglight from the lamp to the apparent mirror of the interferom-eter. Figure 29C details the subsystem in Figure 29A convey-ing light between the apparent mirror of the interferometerand target.

etalons 205 and 207, which are necessary in only one ofthe two interferometers. The gearbox or other meansfor translating the mirrors 217 and 219 in a 2:1 ratio isshown only in one half. The gearbox may also exist onthe illuminating side instead of or in addition to being onthe detecting side. Either gearbox is a convenience andare not required. Some duplicated optics are not labeledfor clarity.

Each half enclosed in the respective dashed box con-sists of three subsystems: a subsystem conveying lightbetween the lamp/camera and the interferometer; theinterferometer; and a subsystem conveying light betweenthe interferometer and the target. Figure 29B shows thefirst subsystem in detail. An important purpose of thissubsystem is to limit the range of ray angles (numeri-cal aperture) and transverse extent of the beam passingthrough the interferometer by means of irises 206 and212, so that none of the light is vignetted by mirrors218, 216 or 220. That is, so that every ray of the beampasses through each of the arms, so that there is no un-equal vignetting of the different paths that would let rays

travel through some paths but not others. This limit-ing can be accomplished either before or after the raystravel through the interferometer. Lens 204 images thelamp/camera 202 to iris 212. Lens 210 images iris 206 tothe A.M. surface 200. Mirrors 208, 214 steer the beam tomake the configuration more compact. Iris 212 controlsthe ray angle range, and iris 206 controls the width ofthe beam at the A.M. 200.

Some kind of numerical aperture limiting systemshould be included with every superimposing interferom-eter described in this document, unless the light source isknown to have a limited numerical aperture less than thesmallest numerical aperture of any interferometer arm.

Figure 29C shows the subsystem conveying light be-tween the interferometer and the target. Lens 222 andlens 226 together image the A.M. 200 to the target sur-face 228. Lens 222 is a field lens to reduce vignettingand should be as close to the A.M. surface as possibleand yet be outside the interferometer cavity (so that itdoesn’t introduce wavefront distortion inside the cavitywhere it is critical). Thus, it lies close to the beamsplitter215. Mirror 224 folds the beam to make the configurationmore compact.

3. Matching dispersion

Adjustable thickness etalons 205 and 207 are used tobalance dispersion between the illuminating and detect-ing interferometers. Only one of the two interferometersneeds to have its dispersion adjusted. One embodimentof a variable thickness etalon is the combination of twoidentical wedge prisms that slide against each other. Afluid filled parallel cavity is another method of creatinga variable thickness etalon. One or more tiltable parallelglass plates at an angle to the beam, is a third method.

The source of the unbalanced dispersion is the variancein thickness of transmissive optics (lenses and beamsplit-ters) that are purchased off-the-shelf. This amount ofunbalanced dispersion is usually equivalent to less than1 millimeter of glass. Since a single thin glass etalon isnot rigid and hard to polish flat, it is better to use twoseparate thick etalons 205 and 207 placed in oppositeinterferometer arms, so that their effects subtract fromeach other. For example, a 5 millimeter etalon and 4.8millimeter etalon placed in opposite arms would have thesame dispersion as a 0.2 millimeter etalon. This allowsadjustable dispersion in the neighborhood of zero to bemade.

It is optimal that the adjustment of the effective etalonlength does not change the angle or position of thebeam from one arm compared to the other, because thatcould degrade alignment of the interferometer. Thus theetalons should be normal to the beam, and if tiltableetalons are used, their effect on the beam position needsto be considered.

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4. Constant-delay mirror motion

To adjust the position of the apparent mirror withoutchanging the interferometer delay, the center mirror 217should be moved without changing the total path lengthfrom mirror 219 to mirror 217 to the beamsplitter 201.This requires moving mirror 217 half the displacementof the displacement of mirror 219. This can be donemanually by incrementing the micrometer screws of thetranslators 211 and 213 in a 2:1 ratio, or done automat-ically by an electrical or mechanical means, such as agearbox or a coordinated pair of stepper motors. Aftermovement of mirrors 217, 219, the field lens 215 is ad-justed to keep the A.M. flat. Since it is a transmissivelens, its adjustment does not change the time delay.

5. Matching procedure

Due to the small (∼1 micron) positional tolerance ofthe delay matching condition τ1 = τ2, the interrelationbetween the mirror positions, time delay, and A.M. posi-tion, the interrelation between the etalon thickness andtime delay, and the weakness of the fringes when not un-der a perfectly matched and aligned state, the process ofmatching the two interferometers can be tedious if thecorrect procedure is not followed. The following proce-dure is recommended:

1. Using string and a ruler, to the nearest millimeter,position the optics of both interferometers to havethe same delay lengths and be at the same calcu-lated positions which will overlap the A.M. withthe short arm mirror. (This adjustment will becalled adjustment of longitudinal overlap.) Set thevariable thickness etalons to have equal thicknessin each arm of the detecting interferometer.

2. Use broad diameter HeNe laser illumination to fo-cus the relay lens systems 230, 218, 220 and 215,217, 219 to each produce a flat A.M. by producingparallel comb of fringes instead of rings.

3. Add neutral density filters if necessary to balancethe intensities between the two arms of each inter-ferometer.

4. By moving the short arm mirrors 216 and 221, op-timize the longitudinal overlap of each interferome-ter by the retro-reflection method described below.Confirm this with the stationary fringe method de-scribed below.

5. The delay τ1 of the illuminating interferometer isnow set. For the remainder of the procedure, theilluminating interferometer optics will not be dis-turbed, only the detecting interferometer will betouched. Search for the matched delay conditionτ1 = τ2, by changing the detecting interferometer

delay until “white light” fringes are seen using ashort coherence length source. The delay can bechanged by movement of the short arm mirror 221.The temporary use of a sodium vapor lamp canhelp localize fringes to a few tenths of a mm. Ini-tially, the fringes may be very weak due to uncom-pensated dispersion. The fringes are most visiblewhen a fringe comb is presented instead of a singleinfinitely wide fringe. A fringe comb is obtainedby slightly tilting one of the mirrors, such as theshort arm mirror 221. This can be done while tem-porarily observing fringes under HeNe illumination,since the HeNe fringes and the eventual white lightfringes will have the same approximate appearance.

6. Null the dispersion. By viewing the white lightfringes through a red color filter, and then througha green color filter, the delay location of the maxi-mum visibility of the red and green fringe compo-nents is noted and compared. That is, scan themicrometer screw which changes the detecting in-terferometer delay τ2 until the red/green fringes areat maximum contrast and note the associated mi-crometer value. When uncompensated dispersionis present, the red and green maxima while lie atdifferent locations. Change one of the etalon thick-nesses slightly and remeasure the red/green max-ima. Keep changing the etalon thickness until thered and green maxima overlap. This will also pro-duce white light fringes with a minimum width andmaximal visibility.

7. Re-optimize the longitudinal overlap in the detect-ing interferometer while keeping the delay constant.The short arm mirror is left fixed and mirrors 217and 219 are moved in a 1:2 ratio to maintain con-stant path length between end mirror and beam-splitter, while changing the inter-mirror (217 to219) separation. A chart analogous to Fig. 8Bshould be calculated to confirm that the configu-ration is not at a stagnant point. Confirm optimaloverlap by the stationary fringe method.

6. Retro-reflecting overlap adjustment subprocedure

The retro-reflecting method of optimizing the longitu-dinal overlap between the A.M. and the short mirror isas follows: A target is temporarily used which stronglyretro-reflects light, such as a bead-painted surface. A50% beamsplitter is inserted into the illuminating beamso that the temporary target can be observed in partialreflection off this beamsplitter. Iris 206 is reduced to asmall pinhole to ease the light load on the eye, since aspecular back-reflection of the lamp of the beamsplitterand mirror 216 will be in the same field of view as thewhite light fringes to be observed. The short arm mirror216 is translated longitudinally several millimeters untilwhite light fringe rings are seen which vary in size as

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a function of mirror position. The mirror is translateduntil the rings become infinite in diameter. This hasempirically been determined to approximately produceoptimal longitudinal overlap between the A.M. with theshort arm mirror. Leave the mirror in this position. Thebeamsplitter and temporary target are then removed andthe iris restored to its original size.

7. Stationary fringe overlap adjustment subprocedure

The stationary fringe method of confirming longitudi-nal overlap is as follows: an approximately collimatedand wide diameter HeNe laser beam is used as a tem-porary source of illumination in place of the white lightbeam from lens 210. A temporary mirror may be in-serted into the path after the iris 212 to introduce thisHeNe beam. This temporary mirror is swiveled by handwhile inspecting the fringe pattern produced on the tar-get. In an aberrationless superimposing interferometer,when the longitudinal overlap is optimum, the fringe pat-tern is stationary despite tilting the angle of incidence ofthe input rays. In the presence of aberrations such asspherical or astigmatic, only the portion of the fringeat the center of the image area may appear stationarywith respect to tilting of the temporary mirror, whilethe periphery changes somewhat. It is optimal that thisstationary portion be as large as possible.

S. Retro-reflecting interferometer as doubleinterferometer

Instead of using two separate interferometer apparati,a single interferometer apparatus can function as a pairby sending light reflected from the target back throughthe same or nearly the same optics. This is called aretro-reflecting configuration, even though the light mayactually travel a slightly different path than true retro-reflection (differing in angle, polarization or position inthe image plane).

The retro-reflecting configuration provides a greatpractical advantage because the delays τ1, τ2 and the dis-persion are then automatically nearly matched, even inthe presence of large mechanical vibration that changesthe delay. Thus this configuration is attractive for use inindustrial settings where vibration is present. Secondly,a rigid optical platform and mounting hardware is not asimportant. This reduces cost and weight. Alignment isgreatly simplified.

There are two kinds of light which must be distin-guished from each other: first passed light and secondpassed light. The first passed light is light going throughthe interferometer for the first time and is used to illu-minate the target. It does not carry any useful velocityinformation about the target. The second passed lightis first passed light after it has reflected from the targetand is going through the interferometer for the second

Fig. 30

Delaying MirrorOut2B

Out1A

= first passed light

= second passed light

Target

Source541

554

540

546

548556

550

Out1B542

543

544

552

Figure 30 shows the use of beam angle to separate doublepassed light from first passed light in a retro-reflecting appli-cation of an interferometer.

time. It is the light which should be detected to pro-duce fringes carrying target velocity information. Thereshould be methods in place which prevent the first passedlight from reaching the detector and being confused withthe second passed light. Because the first and secondpassed light may be sharing portions of the same op-tical path, unwanted reflection from air/glass interfacesor dust on the optics may cause the first passed light toreach the detector. This is a concern, especially since thefirst passed light could be much brighter than the secondpassed light, especially for dark targets. Thus, methodsfor discriminating the two kinds of light are important topractical use of the retro-reflecting configuration.

Since any diffusely reflecting target will produce somelight scattering back toward the source, any single in-terferometer, including single interferometers of a doubleinterferometer system, can be used in a retro-reflectingmode if a means is provided to separate second passedlight from first passed light. This is useful as a diag-nostic tool in aligning each individual interferometer toachieve good superposition, prior to being used as a pair.This method was previously described in the section onpair-matching procedures.

There are several methods of distinguishing the secondpassed light from first passed in a retro-reflecting config-uration: by angle of incidence, by position in the imageplane, and by polarization. These are discussed below.

T. Methods of Discriminating Beams in aRetroreflecting Interferometer

1. Discrimination by beam angle

Figure 30 shows use of beam angle to provide discrim-ination between first and second passed rays. The firstpassed rays are indicated by the open arrowheads and thesecond passed light by the dark arrowheads. Consider

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that for each pass, the interferometer offers two outputpaths. For the first passed light, the outputs are Out1A

540, which heads back approximately toward the source541, and Out1B 542, which heading in the target 543 di-rection. For the second passed light originating at thetarget 543, the two outputs are Out2A heading exactlytoward the source, and Out2B 544 heading only gener-ally toward the target. The mirrors of each arm (shortarm mirror 550 and the apparent mirror 552 of the de-laying mirror assembly 554) are tilted so that light doesnot reflect normally off these mirrors. That is to say, thelight reflecting off the mirrors 550, 552 makes two differ-ent spots 546, 548 where they intersect the beamsplittingsurface 556. This puts the output 544 at a sufficientlydifferent angle from the target beam to allow separationfrom it.

2. Discrimination by polarization and image offset

Figure 31 shows a retro-reflecting superimposing in-terferometer embodiment using two methods of discrim-ination: polarization and image position, which could beused independently. Thirdly, a birefringent wedge is usedto produce a inclined delay for the second passed lightwhile the first passed light sees a uniform delay. Lightfrom source 600 is horizontally polarized by polarizer 602and passes through aperture 604 which defines the extentof the first passed light at the image planes 606, 608, 612and the A.M. 610. These planes are all imaged to eachother by lenses, such as 614, 616. Field lenses analogousto lenses 222 or 204 of Fig. 29A may optimally also beused, but are not shown in Fig. 31 for clarity. The fieldlenses could be placed near the image planes or near thebeamsplitter to reduce vignetting.

A polarizing beamsplitter 601 transmits the firstpassed light, but reflects the second passed light, whichwill be arrange to be polarized in the orthogonal di-rection. The first passed light leaves the interferometerthrough input/output plane 608 to be conducted to thetarget either by an imaging, non-imaging or combinationof imaging and non-imaging systems (618, 622).

Figure 31 shows a combination of imaging and non-imaging systems. Fiber 618 conducts light to the targetimage 628 created by lens 622. Both fibers 618, 620 con-duct light from the target due to intentional slight defo-cusing of the lens 622 between target surface and plane628. However, in this embodiment the exactly retro-reflected light coming back through fiber 618 is not used,since spurious glare from the end of this fiber at plane608 would compete with light from the target. (However,an anti-reflection coating could be used to reduce this.)

Instead, a second fiber 620 is used to conduct light backtoward the interferometer. This way it can be positionedin a separate location in the input/output plane 608 fromthe first passed light beam, so that it can be distinguishedby position. This is done by the use of an aperture 624that is offset in location in reflection of the beamsplitter

601 so that it does not overlap with the aperture 604, sothat aperture 624 blocks first passed light reaching thedetector.

In addition to discrimination by position in the im-age plane, the light can be discriminated by polarization.The fiber 620 is twisted 90 so that there is significantintensity in the vertical polarization. (For long fibers,twisting may be unnecessary because of the polarizationscrambling that occurs). The light then passes througha vertical polarizer to eliminate polarization parallel tothe first passed light. If a purely imaging system is usedwhich is unlikely to scramble or twist the polarization,then a retarder (not shown) can be inserted into the re-flected light beam prior to the polarizer 622 to createsignificant intensity in the orthogonal (vertical) polariza-tion. The polarizing beamsplitter 616 diverts the secondpassed light and not the first passed light toward the de-tector 618. A vertical polarizer offers additional polariza-tion discrimination, in case the polarizing beamsplitter isnot ideal.

U. Birefringent wedge

A birefringent wedge 626 can be used to create aninclined delay for second passed light but not for firstpassed light, because it acts like a wedge for one polar-ization and a different wedge (or a wedge with zero an-gle) for the other polarization. When light travels to thetarget through a non-imaging system (the fiber), theremust be a uniform delay imprinted to the light. Other-wise rays having different imprinted delay values differ-ing more than λ/4 will scramble together, and no fringesmay be produced when the target is subsequently ob-served through an interferometer. However, for the sec-ond passed light an inclined delay is possible, since theinterferometer uses imaging systems so that pixels be-tween the input/output planes are not scrambled.

An inclined delay is useful for determining fringe phaseby recording the fringe comb with a multi-channel de-tector, such as the CCD camera 618. When using aninclined delay, the source of the second passed light atthe plane 608 should optimally be a line source so thereis light at every delay value in a range. This could becreated by use of cylindrical optics (not shown) or a fiberbundle whose component fibers rearrange from a circleto a line. The line direction would be into the Figure,perpendicular to the offset direction.

The birefringent wedge 626 appears as a parallel slabfor first passed light (horizontal polarization) and awedge for second passed light (vertical polarization). Thenet wedge can be constructed of component wedges 401,412, as in Fig. 16, and the net wedge angle and bire-fringent properties adjusted by rotating the componentwedges about the common optic axis.

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BS

Delaying Mirror

Fig. 31

Birefringent wedge

Horiz. Polarizer

Vert. Polarizer

CCD camera ordetector

Target

Input/outputplane

Input/Outputplanes

Fibers

A.M. planeScrambles polarization

Vert. polarizer

= first passed light

= second passed light

602

604606

612

624618

614

616

626

608

622

618

620 610

622

600

628

603

PBS601

605

Figure 31 shows use of offset images to separate double passed light from first passed light in a retro-reflecting application ofan interferometer.

Fig. 16

Optic axis

412410

414

Figure 16 shows the net wedge angle being adjusted by twist-ing component wedges about a common axis.

V. Other offset target image methods

In Fig. 31, distinctly different fibers were used to po-sition input and output light in the image plane 608, sothat first passed and second passed light could be dis-tinguished by position. Figure 32A, B, C shows othermethods of offsetting the incoming from outgoing light,by creating twin images of the target.

Figure 32A shows a prism 650 overlapping a portion

of the lens 652 which images the target 651 to the in-put/output plane 658. This plane could be the in-put/output of the interferometer 608 or the plane 628where the fibers are placed. The latter case is a morelight efficient method of sending light to fiber 620 with-out blurring the lens 622. The prism deviates the anglesso the target appears in two places 656 and 654. Oneplace is used for aiming the illumination, the other foraiming the collection of reflected light. Places 656 and654 are analogous to where fibers 618 and 620 meet theplane 608 in Fig. 31.

Figure 32B shows image offsetting accomplished by asegmented mirror with a nonzero angle θ between the seg-ments 662, 664. This has the advantage over the methodof Fig. 32A in that it is achromatic. (However, the chro-matic blurring of prism 650 may be unimportant if anon-imaging system is used or if the target has a uni-form velocity across its surface.)

Figure 32C shows image offsetting accomplished by asegmented imaging lens 665, where the optical center ofeach lens segment are not co-axial with the others. Thiscould be created conceptually by removing a rectangularsection 666 from a lens and moving the separate segments668, 670 together.

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Reflected Illuminating

Input/outputplane

Fig. 32A

650

652

654 656658

651

Fig. 32B

664

662

Fig. 32C

Remove

665668 670

666

Figure 32A shows use of a prism to create a double image of target to distinguish illuminating from reflected light. Figure32B shows use of a segmented mirror instead of prism in Fig. 32A. Figure 32C shows use of a split lens instead of the prism inFig. 32A.

W. Kinds of waves

For concreteness, light was discussed as the illumina-tion wave kind. However, the interferometer and delay-ing mirror designs can be used with any wave kind whichtravels as rays in 2 or 3 dimensions, such as sound, elec-tromagnetic waves (infrared, ultraviolet, x-rays etc.) ormatter waves (de Broglie waves) provided elements thatfunction analogous to the necessary beamsplitters, lensesand mirrors etc. are used.

Acknowledgments


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IV. CLAIMS

(Not official version. There is a high probability thatthis version is incomplete or differs from the official ver-sion due to many post-submission amendments. Due tothe length and tediousness of the task, I have not retypedthe official version beyond claims 1-7.)

1. An interferometer, comprising:

means for splitting an input beam of light into a first beam oflight and a second beam of light;

means for producing a difference between the apparent and ac-tual path lengths of said first beam using real imaging, whereinsaid difference is called beam shortening;

means for coupling said first beam of light out of said interfer-ometer to produce a first output beam of light; and

means for coupling said second beam of light out of said inter-ferometer to produce a second output beam of light,

wherein said first output beam of light and said second out-put beam of light superimpose to produce a superimposed beam,wherein an object in said input beam seen through the interfer-ometer appears in both images which superimpose longitudinally,transversely and in magnification.

wherein said means for producing a difference between the ap-parent and actual path lengths of said first beam comprise an ap-parent mirror,

wherein said means for producing a difference between the ap-parent and actual path lengths of said first beam comprise an ap-parent mirror, wherein said interferometer further comprises meansfor producing a difference between the apparent and actual pathlengths of said second beam, wherein said means for producing adifference between the apparent and actual path lengths of saidsecond beam comprises a second apparent mirror.

2. The interferometer of claim 1, wherein said first apparent orsaid second apparent mirror is formed by real imaging of an actualmirror.

3. The interferometer of claim 1, wherein the difference betweenthe beam shortening of said first beam and the beam shortening ofsaid second beam is small compared to the absolute value of eitherbeam shortening.

4. The interferometer of claim 1, wherein said apparent mirroris formed by real imaging of a firs actual mirror and said secondapparent mirror is formed by real imaging of a second actual mirror.



means for producing a difference between the apparent and ac-tual path lengths of said first beam using virtual imaging, whereinsaid difference is called beam shortening; wherein said virtual imag-ing is accomplished by a lens and a curved mirror;






means for producing a difference between the apparent and ac-tual path lengths of said first beam using real imaging, whereinsaid difference is called beam shortening;




wherein said means for producing a difference between the ap-parent and actual path lengths of said first beam comprise an ap-parent mirror,

wherein said means for producing a difference between the ap-parent and actual path lengths of said first beam comprise an ap-parent mirror, wherein said beam shortening is produced by imag-ing using a waveguide.



means for producing a difference between the apparent and ac-tual path lengths of said first beam using at least one transparentwedge; wherein said difference is called beam shortening; whereinsaid at least one transparent wedge creates a beam shortening thatvaries across said first beam;




9. The interferometer of claim 7, wherein said at least one trans-parent wedge comprises a plurality of wedges, wherein each wedgeof said plurality of wedges has a unique dispersive power with re-spect to the other wedges of said plurality of wedges, wherein thethickness of each wedge of said plurality of wedges is chosen so thatthe slope of said beam shortening versus location across said firstbeam is achromatic.

11. The interferometer of claim 7, wherein an etalon is placedin said second beam, wherein the thickness and dispersion of saidetalon are chosen to compensate the wavelength dependence of thebeam shortening created by said wedge.

13. The interferometer of claim 7, wherein said at least onetransparent wedge comprises birefringent properties; wherein theslope of the beam shortening for one polarization of light may bedifferent than for the orthogonal polarization; wherein a slope ofbeam shortening is a rate of change of beam shortening versusposition across a beam;

16. The interferometer of claim 3, wherein the difference betweenthe beam shortening of said first beam and the beam shortening ofsaid second beam is small compared to the absolute value of eitherbeam shortening.

17. The interferometer of claim 3, wherein said apparent mirroris formed by real imaging of a first actual mirror and said secondapparent mirror is formed by real imaging of a second actual mirror.



means for producing a difference between the apparent and ac-tual path lengths of said first beam using a first etalon, whereinsaid difference is called first beam shortening;

means for producing a difference between the apparent andactual path lengths of said second beam using a second etalon,wherein said difference is called second beam shortening; whereinthe refractive properties of said second etalon over the operationalwavelength range of said interferometer differ from refractive prop-erties of said first etalon;


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20. An interferometer, comprising:means for splitting an input beam of light into a first beam of

light and a second beam of light;means for producing a difference between the apparent and ac-

tual path lengths of said first beam wherein said difference is calledbeam shortening;

means for beamsplitting inserted into either said first beam orsaid second beam to create a finite number of additional splitbeams; wherein said split beams are coupled out of said interfer-ometer to produce additional output beams;



wherein said first output beam of light and said second outputbeam of light and said additional output beams superimpose to pro-duce a superimposed beam, wherein an object in said input beamseen through the interferometer appears in multiple images whichsuperimpose longitudinally, transversely and in magnification.

21. The interferometer of claim 20, wherein said at least onemeans for beamsplitting are polarization dependent, wherein thepolarization state among said said first output beam of light andsaid second output beam of light and said additional output beamsmay differ.

22. The interferometer of claim 1, wherein at least one trans-parent lens of positive focal length is used to produce said meansfor producing a difference between the apparent and actual pathlengths of said first beam using real imaging, wherein said firstbeam makes a circulating path so that it periodically reaches saidmeans for coupling said first beam of light out of said interferom-eter, wherein for a given input pulse entering said interferometeralong said input beam, said second output beam and said first out-put beam comprise an infinite number of geometrically decreasingintensity output pulses that leave said interferometer.

23. The interferometer of claim 22, wherein said circulatingpath is created between a first end mirror and a second end mir-ror, wherein an imaging system lies between said first end mirrorand said second end mirror; wherein the surface of said first endmirror forms a real image on the surface of said second end mirror.Wherein the center of curvature of said first end mirror is imagedto the center of curvature of said second mirror.

24. An interferometer, comprising:means for splitting an input beam of light into a first beam of

light and a second beam of light;means for producing a difference between the apparent and ac-

tual path lengths of said first beam, wherein said difference is calledbeam shortening;



wherein said first output beam of light and said second out-put beam of light superimpose to produce a superimposed beam,wherein an object in said input beam seen through the interfer-ometer appears in both images which superimpose longitudinally,transversely and in magnification; and

means for separating into distinct paths light incident upon atarget from light returning from said target when said target isplaced in the superimposed path of said first output beam and saidsecond output beam; wherein said means for separating includes anoptical system that creates a double image of a target, wherein eachimage of said double image of said target are transversely shiftedas they appear to said interferometer output beam.



means for producing a difference between the apparent and ac-tual path lengths of said first beam, wherein said difference is calledbeam shortening;

means for adjusting the amount of dispersion in said first beamby varying the pathlength of said first beam through a transparentmedium having a dispersive refractive index;




30. The interferometer of claim 1, further comprising meansfor moving said imaging elements that affect said beam shorteningin a coordinated fashion, wherein said coordination is intended tochange primarily one parameter while minimizing change of otherparameters, wherein said one parameter includes said actual pathlength, said apparent path length, beam wavefront curvature, im-age magnification, and tranverse superposition, wherein said beamwavefront curvature is the center of curvature of the wavefront ofsaid beam when plane wavefronts are incident on said interferome-ter; wherein said means for producing a difference between the ap-parent and actual path lengths of said first beam are called imagingelements;






wherein said first output beam of light and said second out-put beam of light superimpose to produce a superimposed beam,wherein an object in said input beam seen through the interfer-ometer appears in both images which superimpose longitudinally,transversely and in magnification; and

means for chromatically dispersing said superimposed beam,wherein said means for chromatically dispersing include means fordirecting said superimposed beam into multiple channels organizedby wavelength.


means for chromatically dispersing an input beam into an inputspectrum;

means for splitting said input spectrum into a first beam of lightand a second beam of light;




wherein said first output beam of light and said second out-put beam of light superimpose to produce a superimposed beam,wherein an object in said input spectrum seen through the interfer-ometer appears in both images which superimpose longitudinally,transversely and in magnification.

35. A double interferometer, comprising:

a source of illumination;

a first interferometer comprising:

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wherein said first output beam of light and said second out-put beam of light superimpose to produce a superimposed beam,wherein an object in said input beam seen through the interfer-ometer appears in both images which superimpose longitudinally,transversely and in magnification, wherein delay between said firstoutput beam and said second output beam is called delay 1;

a second interferometer comprising:second means for splitting a second input beam of light into a

third beam of light and a fourth beam of light, wherein said secondinput beam of light comprises said superimposed beam, whereinthe distance between said means for coupling said second beam oflight out of said interferometer to said second means for splittingsaid second input beam is defined as optical path;

second means for producing a difference between the apparentand actual path lengths of said third beam, wherein said differenceis called beam shortening;

second means for coupling said third beam of light out of saidsecond interferometer to produce a third output beam of light; and

second means for coupling said fourth beam of light out of saidsecond interferometer to produce a fourth output beam of light,

wherein said third output beam of light and said fourth outputbeam of light superimpose to produce a second superimposed beam,wherein a second object in said second input beam seen throughsaid second interferometer appears in both images which superim-pose longitudinally, transversely and in magnification wherein de-lay between said third output beam and said fourth output beamis called delay 2; wherein said delay 2 approximately matches saiddelay 1 so that fringes are observed in second superimposed beam,

wherein said said superimposed beam is directed to and reflectedfrom a target before said superimposed beam enters said secondinterferometer, wherein time rate changes in length of said opticalpath cause a shift in the phase of said fringes, wherein the lengthof said optical path is the integration of the refractive index alongphysical path of said optical path, wherein said time rate changes ofoptical path include motion of said target and changes in dielectricproperties of said target,

wherein communication is transmitted from said first interfer-ometer to said second interferometer by modulating said delay 1,which changes appearance of said fringes in said second interfer-ometer.

36. A method of matching delay and dispersion of two interfer-ometers of a double interferometer, comprising:

setting a first delay of a first interferometer and second delay ofa second interferometer to be the same within a few milllimetersby initial positioning of optical elements of said first interferometerand said second interferometer, wherein said first delay is calleddelay 1 and said second delay is called delay 2;

maximizing the longitudinal overlap of said first interferometer,maximizing the longitudinal overlap of said second interferome-

ter, adjusting said delay 2 to match said delay 1, wherein a short co-

herence length illumination source is used to send a beam throughsaid first interferometer which then passes through said second in-terferometer, wherein the visibility of fringes seen in said secondinterferometer are maximized while said delay 2 is adjusted bymovement of optical elements of said second interferometer;

matching dispersion between said first interferometer and saidsecond interferometer; wherein at least one variable thicknessetalon is adjusted in thickness to minimize difference in locationof the center of fringes for two different wavelengths of illumina-tion, wherein said delay 2 is held constant; and

maximizing longitudinal overlap of said second interferometerusing a retro-reflective configuration, further comprising holdingsaid delay 2 constant, wherein positions of optical elements of saidsecond interferometer are moved in a coordinated schedule calcu-lated to hold said delay 2 constant while adjusting said longitudinaloverlap of said second interferometer.

41. A delay line comprising:

an input beam;

a beamsplitter which redirects some or all of said input beam oflight into a new direction to produce a first beam of light;

means for producing a difference between the apparent and ac-tual path lengths of said first beam of light using real imaging of amirror, wherein said first beam of light is reflected from said appar-ent mirror to form a reflected beam; wherein said reflected beamhas a path distinct from said input beam.

42. The delay line of claim 41, wherein said beamsplitter is po-larization dependent; wherein said beamsplitter has enhanced re-flectance for a specific state of polarization of said input beam andenhanced transmission for the complementary polarization state;further comprising a means for changing polarization of said re-flected beam to enhance in said reflected beam the amount of in-tensity in said complementary polarization state.

43. The interferometer of claim 7, further comprising a meansfor producing a difference between the apparent and actual pathlengths of said second beam.

44. The interferometer of claim 43, wherein said means for pro-ducing a difference between the apparent and actual path lengthsof said second beam uses real imaging.

45. The interferometer of claim 43, wherein said means for pro-ducing a difference between the apparent and actual path lengthsof said second beam uses virtual imaging.

46. The interferometer of claim 7, further comprising a secondmeans for producing a difference between the apparent and actualpath lengths of said first beam.

47. The interferometer of claim 29, wherein said means for ad-justing the amount of dispersion comprises at least one variablethickness etalon.

48. The interferometer of claim 29, wherein said means for ad-justing the amount of dispersion comprises at least one elementselected from a group of wedge and etalon; wherein orientationof said at least one element relative to said first beam is changedthereby producing a changing pathlength through said at least oneelement.

49. The interferometer of claim 29, wherein said means for ad-justing the amount of dispersion comprises a plurality of wedges;wherein orientation of component wedges of said plurality of wedgesrelative to each other is changed thereby producing a changingpathlength through said plurality.

IL-10168FringingSpec13.tex 12/03

Part 6

CombinedDispersive/InterferenceSpectroscopy forProducing a VectorSpectrum

US Patent 6,351,307Issued Feb. 26, 2002


A method of measuring the spectral properties of broadband waves that combines interferometrywith a wavelength disperser having many spectral channels to produce a fringing spectrum. Spectralmapping, Doppler shifts, metrology of angles, distances and secondary effects such as temperature,pressure, and acceleration which change an interferometer cavity length can be measured accuratelyby a compact instrument using broadband illumination. Broadband illumination avoids the fringeskip ambiguities of monochromatic waves. The interferometer provides arbitrarily high spectralresolution, simple instrument response, compactness, low cost, high field of view and high efficiency.The inclusion of a disperser increases fringe visibility and signal to noise ratio over an interferometerused alone for broadband waves. The fringing spectrum is represented as a wavelength dependent2-d vector, which describes the fringe amplitude and phase. Vector mathematics such as gener-alized dot products rapidly computes average broadband phase shifts to high accuracy. A Moireeffect between the interferometer’s sinusoidal transmission and the illumination heterodynes highresolution spectral detail to low spectral detail, allowing the use of a low resolution disperser. Mul-tiple parallel interferometer cavities of fixed delay allow the instantaneous mapping of a spectrum,with an instrument more compact for the same spectral resolution than a conventional dispersivespectrometer, and not requiring a scanning delay.

I. INTRODUCTION


The present invention relates to interferometric anddispersive spectroscopy of broadband waves such as light,and more specifically the interferometric measurement ofeffects which can be made to produce phase shifts such asDoppler velocities, distances and angles, and furthermorethe mapping of spectra.

B. Description of Related Art: Spectroscopy

Spectroscopy is the art of measuring the wavelength orfrequency characteristics. There are two complementaryforms of spectroscopy method currently used today.


Dispersive Spectrographs In the oldest form, a prismor grating disperses input illumination (let us call itlight) into independent channels organized by wave-length or frequency. A spectrum is created whichis the intensity versus wavelength channel. This isa scalar versus wavelength channel.

Interferometric Spectrographs In the other method,called Fourier transform spectroscopy, an interfer-ometer having a variable path length difference(called the delay) interferes the illumination witha delayed copy of itself, creating an interferogram.The Fourier transform of this yields the spectrum.Previously, the two methods have not been used to-gether where the interferometry and dispersivenesshave had equal emphasis.

1. Doppler Velocimetry

An important practical use of spectroscopy is the mea-surement of Doppler shifts. In addition to many indus-trial applications of Doppler velocimetry, astronomers

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I2 vapor

CCD

ImagingDisperser

PZT

LampStar

Arm A

Arm B

Slitdirection

Fiber

dummy

1 3

5

7

911

13

15

17

19

2123

25

27 29 31

33

35

37

39

41

43

Figure 1

Interferometer

Figure 1A shows an embodiment that measures Doppler shifts of starlight. Figure 1B shows a fringing spectrum or 2-dimensionalintensity image produced by the embodiment of Figure 1A.

measure the Doppler velocity of stars in order to deducethe presence of planets orbiting around the star. Thestellar spectrum contains numerous dark absorption linesagainst a bright continuum background. These spectrallines are randomly distributed about 1 A apart from eachother. A slight change in the average position of theselines is the Doppler effect to be measured. The averagewidth of these stellar lines is about 0.12 A in the visi-ble, which corresponds to an equivalent Doppler velocitywidth of about 6000 m/s. Hence, measuring Dopplervelocities below 6000 m/s is extremely challenging andrequires carefully dividing out the intrinsic behavior ofthe instrument from the raw data.

A 1 m/s velocity resolution is desired in order to reli-ably detect the presence of Jupiter and Saturn-like plan-ets, which produce 12 m/s and 3 m/s changes respec-tively in the stars intrinsic velocity. Current astronom-ical spectrometers are based on the diffraction grating.These have a best velocity resolution of 3 m/s, but is of-ten 10 m/s in practice. This resolution is insufficient toreliably detect Saturn-like extrasolar planets. This limitis related to the difficulty in controlling or calibrating thepoint spread function (PSF).

2. Drawbacks of Purely Dispersive Spectrosgraphs

The PSF is the shape of the spectrum for a perfectlymonochromatic input. Ideally this is a narrow peak of

well-determined shape. Unfortunately, the PSF of ac-tual gratings varies significantly and in a complicatedway against many parameters such as temperature, time,and average position in the spectrum. It is a complicatedfunction that requires many mathematical terms to ad-equately approximate it. This is fundamentally due tothe hundreds or thousands of degrees of freedom of thediffraction grating– at least one degree of freedom pergroove of the grating. These degrees of freedom mustbe carefully calibrated, otherwise drifts can cause appar-ent Doppler velocities much larger than the effect beingsought. The calibration process is time consuming.

Another disadvantage of conventional astronomicalspectrometers is their large size, which can be severalmeters in length. Large distances between optical com-ponents, which need to be held to optical tolerances, re-quire very heavy and expensive mounts and platforms toprevent flexure. This dramatically increases expense andprevents portability. Practical use aboard spacecraft oraircraft is prevented. The high expense limits the num-ber of spectrometers which can be built to a few, only bywell-endowed institutions.

Other disadvantages include a very limited field ofview, which is called etendue and is the area of the inputbeam times its solid angle. This is due to the narrow-ness of the slit at the instrument entrance that definesthe range of entry angles. In a grating or prism based in-strument the entry angle and the wavelength, and hencededuced Doppler velocity, are directly linked. The slit

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needs to be narrow to provide better than 0.05 A reso-lution to resolve the stellar spectral lines. Atmosphericturbulence causes the star image to dance around, some-times off the slit opening. This reduces the effective in-strument throughput. Furthermore, changes in intensityprofile across the slit have to be carefully deconvolvedfrom the data, since the Doppler velocity gradient acrossthe slit is approximately 3000 m/s. Thus achieving 3 m/svelocity accuracy is extremely difficult with a dispersivespectrometer, and 1 m/s has never been achieved.

3. Drawbacks of Purely Interferometric Spectrosgraphs

An interferometer is attractive for spectroscopy be-cause its angular dependence can be made very smallor zero. This allows wider slits, and hence accommodat-ing blurrier star images at high throughput, for the sameequivalent spectral resolution. Secondly, its PSF is a si-nusoid, which is a simple mathematical function havingonly 3 degrees of freedom (phase, amplitude and inten-sity offset). This makes calibration of instrument andprocessing of data fast, since standard vector mathemat-ics can be used. Secondly, this makes it easy to rejectnoise not having the expected sinusoidal shape. Further-more, the spectral resolution can be made almost arbi-trarily large simply by increasing the delay (difference inpath length between the two interferometer arms). Theinterferometer is compact and inexpensive, because theoptical components need only be a few millimeters orcentimeters from each other.

The important difficulty of an interferometer mea-suring broadband illumination is poor fringe visibility.Fringe phases naturally changes with wavelength. Whencomponent fringes of many wavelengths combine on thesame detector, they reduce the visibility of the net fringe.For this reason conventional interferometer based instru-ments such as Fourier transform spectrometers withoutany wavelength restricting filters are rarely used in lowlight applications.

4. Prior Dispersive Interferometers

A solution to this problem is to combine a wavelengthdisperser with the interferometer so that fringes of dif-ferent wavelengths do not fall on the same place on thedetector. The combination of disperser and Fabry-Perotinterferometer is described in the book “Principles of Op-tics” by Max Born and Emil Wolf, Pergamon Press, 6thedition, on page 336, section 7.6.4 and their Figure 7.63,7.66 and references therein. Distinctions exist betweenapparatus described in “Principles of Optics” and thepresent invention. The Born & Wolf device producesfringes that are narrow and peak-like, not sinusoidal.Consequently, the fringe shape is not described by a 2-element vector. This reduces accuracy when trying tomeasure small phase shifts. Furthermore, phase stepping

is not involved. Thirdly, a heterodyning action is notemployed to shift high-resolution spectral details to lowspectral resolution.

5. Prior Non-dispersed Interferometer for Metrology

In some kinds of metrology a secondary effect, such astemperature, pressure or acceleration, is measured by thechange it induces in the delay of an interferometer, suchas through changing the position of a reflective surface oraltering a refractive index. The delay is then sensed bythe phase of a fringe. In current devices monochromaticillumination, such as laser illumination, is needed to pro-duce visible fringes from non-zero delays. (The delays areoften non-zero for practical reasons, or to have a signifi-cant range of travel.) However, the use of monochromaticillumination creates fringe skip ambiguities which makethe absolute size of the effect being measured ambiguous.Only small changes can be reliably measured. Broadbandlight solves the fringe skip problem, but produces insuffi-cient fringe visibility because its coherence length (about1 micron) is usually very much shorter than the delay.

6. Prior use of Interferometry for Angle Measurement

In a related metrology, fringe shifts can be used tomeasure angles of distant objects such as stars. Lightfrom the star is collected at two separate places a base-line distance apart, and interfered against each other ata beamsplitter. This is called long baseline interferome-try. Effectively, an interferometer is created in the trian-gle consisting of the target and the two collecting ports.In the case of broadband targets such as starlight, theshort coherence length of the illumination (about 1 mi-cron) restricts the interferometer delay to be very nearzero in order to produce visible fringes. This restrictsthe angular range. Secondly, an interferometer’s phase issensitive to the illumination’s spectral character on band-width scales given by 1/(delay). Having a small or zerodelay means the interferometer phase is sensitive to theoverall shape of the illumination spectrum, and this canvary erratically due to atmospheric turbulence. Moreaccuracy could result if the interferometer were only sen-sitive to behavior on short bandwidth scales, such as 1A, because this is less affected by the atmosphere. Thiswould require using large delays (several millimeters atleast), which can’t be done with the present long baselineinterferometers.

A related spectroscopic long-baseline interferometertechnique is described by Kandpal et. al., in Journalof Modern Optics, vol. 42, p447-454 (1995). Intensitymodulations are observed in a spectrometer which aredue to the angular separation of two stars. However, thistechnique does not use heterodyning, nor phase steppingnor slit fringes, nor an iodine cell, nor use vectors to de-scribe the data at each wavelength channel. Because of

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this, the typical maximum angle it can measure is 8000times smaller than what my invention can measure.


A. Combine Dispersion with Interference

It is an object of the invention to measure the spec-tral characteristics of waves, especially broadband waves.These waves include electromagnetic waves, and anyother waves that can be passed through an interferometerwhere they interfere with a delayed copy of themselves,and can be dispersed into intensity-detecting channels or-ganized by frequency or wavelength. The dispersion canbe either before or after the interference. Broadbanded-ness could be defined as when, with the interferometerwere used by itself without the disperser, the phases ofthe fringes of different spectral regions within the inputillumination are more than 90different from each other,and thereby start to diminish the net fringe visibility.

The invention comprises the series combination of adisperser which organizes the waves by frequency orwavelength, and the interference of the waves with adelayed copy of themselves. It is an object of the in-vention to create a spectrum which has fringes whosephase and amplitude can be determined for a given wave-length channel independent of information from otherwavelength channels. Such a spectrum is called a fringingspectrum.

B. Spatial or Temporal Phase Stepping

To determine fringe phase and amplitude of a givenwavelength channel independent of other channels, theinterferometer delay is arranged to vary, either spatiallyalong the slit of the disperser (which is perpendicular tothe dispersion axis), or temporally by “phase stepping”,which is to take repeated exposures while changing theoverall interferometer delay for all channels. When thedelay changes along the slit, such as by tilting an in-terferometer mirror or beamsplitter, fringes are createdwhich cause the intensity profile along the slit to varysinusoidally with a finite period.

Regardless of any fringing behavior along the slit di-rection, an interferometer always has sinusoidal behav-ior versus frequency. This is called the “spectral comb”.This comb may not be resolved by the disperser, but itspresence is still key in producing Moire fringes.

C. A Heterodyning Effect

The effect of passing light through both the interfer-ometer and disperser is to multiply the spectral combwith the illumination spectrum. Together with the pres-ence of blurring along the dispersion axis, a heterodyn-

ing effect occurs which creates Moire fringes. These shifthigh spectral resolution details to low spectral resolu-tion. Hence, a low spectral resolution disperser can beused, even though high spectral resolution informationis being sensed. This lower costs, increases throughputand increases field of view compared to a high-resolutiondisperser used alone.

D. Vector Spectra

It is an object of the invention to express the fringingspectrum as a 2-dimensional vector versus wavelengthor frequency channel, which is called a vector spectrum.This data format is also called a “whirl”. The length andangle of the vector when expressed in polar coordinatesrepresent the fringe amplitude and phase, respectively.The vectors can be computed by evaluating the Fouriersine and cosine amplitudes, for a periodicity near the nat-ural fringe periodicity along the slit axis, and assigningthese to the X and Y rectangular coordinates of the vec-tor. In the case of infinite fringe periodicity along theslit, the Fourier components cannot be determined froma single exposure, but can be determined if the severalexposures are made while incrementing the interferome-ter by a small amount, such as equivalent to a quarterwave, and knowing that the fringes will shift in phase pro-portional to the delay change. This technique is calledphase stepping. Phase stepping is recommended even forfinite fringe periodicity along the slit, because it assistsin distinguishing true fringes from common-mode noise.

E. Use for Doppler Velocimetry

The embodiment of the invention having a singleapproximately fixed interferometer delay can measurebroadband phase shifts due to the Doppler effect of amoving source. Due to the action of the disperser, theoptimum delay value is approximately half the coherencelength (λ2/∆λ) of the illumination that is due to thespectral lines or other narrow features, not the short co-herence length due to the continuum background. Thatis, ∆λ is given by the 0.12 A width of the spectral lineinstead of hundreds of A of the continuum. In the ab-sence of a disperser the relevant coherence length wouldbe due to the broad continuum background, and there-fore thousands of times shorter.

This delay choice provides a good tradeoff betweenfringe visibility and phase shift per velocity ratio. Forstarlight this is a delay of about 11 mm. The Dopplervelocity is proportional to the whirl rotation. This canbe found by taking the dot product of the input whirlagainst an earlier measured whirl, and against the earliermeasured whirl rotated by 90. The whirl dot productsare generalized dot products evaluated by summing oraveraging the channel dot product over all wavelengthchannels. The dot product is called “generalized” be-

6-5

cause it sums products over both the spatial and wave-length indices. The subsequent arctangent of the twoaforementioned dot products yields the whirl angle. Notethat a key advantage implicit in the generalized dot prod-uct is that the summation over wavelength channels hap-pens prior to applying the arctangent function. This pre-vents large discontinuities in the arctangent function thatwould occur for spectral channels that have zero or smallfringe visibility.

Since the whirl rotation is dependent on both the inter-ferometer delay and the illumination spectrum, measur-ing a Doppler effect requires independently determiningthe interferometer delay, which could be wandering dueto vibration and thermal drift. This can be accomplishedby including a reference spectrum with the target illumi-nation, such as by passing the light through an iodinevapor cell which imprints its own absorption lines, whichhave stable positions unrelated to the Doppler effect.This creates a net whirl which contains two components,corresponding to the target illumination and the refer-ence spectrum. The difference in rotation between thetarget whirl and the reference whirl components yieldsthe target Doppler velocity. The rotational positions ofthe target and reference whirl components can be foundby expressing the total whirl as a linear combination ofcomponent whirls with unknown coefficients, and apply-ing generalized dot products.

The advantage of this invention is an instrument whichis much more compact, lower cost, and having a greaterfield of view than a conventional dispersive spectrometerof 0.05 A, and has much greater signal to noise ratio thanan interferometer used alone.

F. Use for Metrology of Secondary Effects

Another embodiment of the invention uses the interfer-ometer delay as a means of measuring secondary effectssuch as temperature, pressure and acceleration. Theseeffects are arranged to change the delay in a known man-ner, such as by moving a reflective part of the interferom-eter cavity or altering the refractive index of the cavitymedium. A steady reference spectrum is used for the il-lumination. Then changes in the whirl rotation can beascribed to changes in the interferometer delay and hencethe secondary effect. Since the interferometer delay mayalready be changing due to deliberate phase stepping,it is advantageous to include a second interferometer inseries with the probe interferometer cavity to act as areference cavity. Then the change of one interferometerdelay compared to the other provides the measurementof the secondary effect being probed. This method differsfrom conventional interferometric measurements of cav-ity lengths by the use of broadband instead of monochro-matic illumination. This allows a unique determinationof the absolute cavity length without the fringe-skip am-biguity of monochromatic waves.

G. Use for Measuring Angles

A variation of this method can measure the angular po-sition of distant objects such as stars if the interferometercavity is replaced by a long baseline interferometer, whichcollects starlight at two separate places a baseline apartfrom each other and interferes them. Effectively, the tri-angle consisting of the star and the two collection portsforms the interferometer cavity. Changes in star anglecause an arrival time difference in the starlight interferingwith itself. This creates a rotation of the whirl which ismeasured. The target light can be passed through an io-dine vapor absorption cell to imprint a known spectrum.This way, the star’s velocity will not affect the measure-ment because the stellar spectrum can be ignored, andtargets having no intrinsic spectral lines or narrow fea-tures can be used. Furthermore, greater angular separa-tions can be measured because the iodine linewidths aremuch narrower (by a factor 8) than stellar linewidths,so proportionately greater arrival time differences at thebeamsplitter can be measured. By observing two or threestars simultaneously through the use of beamsplitters atthe collection ports, poorly known instrumental param-eters such as the baseline length and the phase steppingamount can be determined from the data. The advan-tage of using this technique is that more precise phaseshifts can be measured than in conventional long base-line interferometry, over larger angular range, becausethe imprinted iodine spectrum is much more informationrich than the star’s intrinsic spectrum.

Furthermore, the 2-d vector format of the whirl pre-serves the polarity of the Moire fringes, which preservespolarity of arrival time difference and hence the targetangle relative to the baseline. This is not possible withdevices that only record the intensity (scalar) spectrum.

H. Use for Mapping a Spectrum

Measuring a Doppler shift of a spectrum is a differenttask than mapping a spectrum, which is the purpose ofmany conventional spectrometers. The former task re-quires only a subset of the full spectral information. Anembodiment of the invention which modifies the interfer-ometer to have several parallel channels of different de-lay striking separate positions of the detector can indeedmap a spectrum. Each interferometer delay produces awhirl which is responsible for measuring a particular sub-set of the full spectral information. The interferometerdelays are chosen to be different from each other so thatthe set of them samples all the desired spectral infor-mation. The separate spectral information is combinedtogether mathematically, using the knowledge that theMoire fringes of the fringing spectrum manifest a het-erodyning process that beats high spectral detail to lowspectral detail. The reverse process is employed to math-ematically reconstruct the full spectrum of the input il-lumination. This process involves Fourier transforming

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each whirl from frequency-space to delay-space, trans-lating each in delay-space by the specific interferometerdelay used, concatenating all these together, then inverseFourier transforming to yield a net spectrum. The ad-vantage of this invention for mapping a spectrum is thatit forms a more compact, lower cost device than disper-sive spectrometers for the same equivalent spectral reso-lution. And compared to conventional Fourier transformspectrometers it can measure single shot events (instanta-neous measurement) because it does not require waitingfor an interferometer delay to be scanned. The delaysare fixed. The lack of significant moving parts is also anadvantage for space-based operation.


A. An Embodiment For Doppler Velocimetry

Figure 1A shows an embodiment of the invention opti-mized to detect small Doppler shifts from starlight. Theseries combination of an interferometer 17 and a disperser35 creates a fringing spectrum 39, shown in Figure 1B,which is a 2-dimensional intensity image. The dimen-sions are direction along a slit 33 versus dispersion axis(frequency or wavelength). This image is recorded at theCCD detector 37 or similar 2-dimensional intensity de-tector. In general the interferometer can either precedeor follow the disperser, since the transmission behaviorof the net instrument is the product of multiplying thetwo transmission behaviors of its components. The slit33 which defines the entrance to the disperser 35 is per-pendicular to the dispersion direction, which is in theplane of the paper. When the interferometer precedesthe disperser, the disperser should be an imaging dis-perser, whereby the intensity variations along the slit di-rection are faithfully imaged to the detector. Otherwise,the apparatus must be operated in mode of infinitely tallfringe period, which relies on phase-stepping to deter-mine fringe phase and amplitude. If the disperser pre-cedes the interferometer, then it is not necessary for thedisperser to preserve the image along the slit.

The choices of the source of light may include a target1, or a lamp 3 having a broad featureless spectrum (suchas an incandescent lamp). The light can conveniently beconducted to the apparatus via a fiber 5. The lamp isneeded during calibration procedures that measure theiodine cell 7 spectrum by itself, or the interferometer byitself.

1. Spectral Reference

The iodine vapor cell 7 is used to imprint a refer-ence spectrum on the input illumination so that unavoid-able changes in interferometer delay such as due to ther-mal drift, air pressure changes, mechanical vibrations

52

50

Intensity

Along slit

a)

56

54

b)Fig. 2

Figure 2a and 2b show a ladder of fringes overlaying the slit.

0° 90° 180° 270°c)

0° 90° 180° 270°d)

Figure 2c and 2d show how the ladder cycles versus fringephase.

and uncertainties in the PZT transducer can be distin-guished from legitimate changes in fringe phase due tothe Doppler effect. These delay drifts can easily exceedthe desired instrument phase accuracy by many ordersof magnitude. For example, measuring a 1 m/s Dopplershift with an 11 mm delay corresponds to a phase changeof 1/14000th of a wave of green light.

The spectral reference should have many narrow andstable spectral lines distributed evenly over the spectrum.The spectral reference can be absorptive, such as iodineor bromine molecular vapors, or emissive such as a tho-rium lamp. An iodine vapor cell heated to 40 to 50 C tem-perature is a common spectral reference used for greenlight. Since absorption is a multiplicative effect, the cellcan be placed anywhere upstream of the detector. Anidentical but empty dummy cell 9 can be swapped withthe iodine cell 7 to reduce the change in beam profiledue to the glass windows. If a reference emission lamp isused, its light should join the target light prior to the in-terferometer and disperser, and be superimposed in pathsuch as by use of a beamsplitting surface.

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2. Optics Preparing Beam

An optical system represented by lenses 13 and 15 im-ages the light from the fiber end 11 through the inter-ferometer and to the disperser slit 33. Other lenses suchas field lenses are not shown but can be included to helprelay the beam and control the pupil from expanding toolarge. These field lenses could image the pupil at lens13 to the pupil at lens 15, for example. For maximumthroughput the lens magnifications are chosen so thatthe size and numerical aperture of the light leaving thefiber is matched to the slit width and accepting numer-ical aperture of the disperser. It is useful to first imagethe fiber end 11 to a real image near the interferometermirror 29 and 25, then to relay this to another real imageat the slit. This minimizes the area of the beamsplitterwhich needs to be optically flat.

It is optimal to make several fringe periods span alongthe slit length, so that fringe phase and amplitude canbe determined for each wavelength channel independentof others in a single exposure. This requires illuminat-ing a length of the slit. To avoid wasting light, the beamcross-section at the slit should be rectangular with a highaspect ratio so that it is not unnecessarily wider than theslit. The beam should also be rectangular at the interfer-ometer mirrors 25, 29 so that a fringe ladder 52 and 54(Fig. 2a and 2b) can be imprinted on it. In applicationswhere efficiency is not important, it is sufficient to useround beam cross-sections.

There are several methods of creating a rectangularbeam cross-section. One method is to use cylindrical op-tics. The optical system represented by lens 13 betweenthe fiber 5 and interferometer 17 could be astigmatic sothat for the plane out of the paper the light would not fo-cus at the mirrors 25, 29 but instead have a large extent.In the plane of the paper it would form a narrow focusat the mirrors 25, 29. Another method is to use a fiberbundle to change a round cross-section to a rectangularcross-section at position 11. Then the optical system rep-resented by lens 13 could be ordinary (not astigmatic). Athird method is to use various “image slicing” techniquesdeveloped by astronomers using mirrors etc. to dissect around beam cross-section and reassemble it in a differentorder to form a rectangle at position 11. If the disperserprecedes the interferometer, then the beam entering thedisperser could be small and round. The disperser couldmake a spectrum which could have finite vertical height(the dispersion direction assumed horizontal), possiblyusing either non-imaging paths inside the disperser orusing cylindrical optics after the disperser and before theinterferometer.

B. The Interferometer

The interferometer 17 is responsible for creating asinusoidal-like frequency dependence to the transmittedintensity. These fringes in frequency space are called

frequency

slit

dire

ctio

n

a) 63

6765

64

frequencysl

it di

rect

ion

b)66

69

Fig. 3

Figure 3a shows a slanted spectral fringe comb created byinterferometer when a ladder of fringes exists along the slit.Figure 3b shows a nonslanted spectral fringe comb created byinterferometer when a single tall fringe overlays the slit.

“spectral comb” fringes, shown as 67, 69 in Fig. 3a and3b. In addition, it is optimal to have spatial fringes 65along the slit direction, so that without phase steppingthe fringe phase and amplitude can be determined fora given wavelength channel. That way the inclusion ofphase stepping results in even better accuracy becausethen there are redundant methods of determining fringephase and amplitude.

A variety of interferometer types can be used, with thedesirable properties being high throughput, and the cre-ation of fringes with a large sinusoidal component. AMichelson type interferometer, as opposed to a high-finesse Fabry-Perot type, creates the most sinusoidalfringes. The optical path length difference between theinterferometer arms is called the “delay”, which is usu-ally in length units. The symbol τ represents the delayin time units,

τ = (delay)/c, (1)

where c is the speed of light.

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c)

FWHM520

half-period

504Bright half

Stellar spectrum

Intensity

frequency

506 Interferometer spectrum505

503

Figure 3c shows that this delay choice is equivalent to makingthe half-period 500 of the comb fringes 505 about equal tothe typical FWHM linewidth 502 of features in the stellarspectrum 503.

1. Fringes in Frequency and/or Space

A Michelson with a non-zero delay will produce a si-nusoidal transmitted intensity component of the formsin (2πfτ) or sin(2πτc/λ). Hence there are three waysthat fringes can be made: 1) Versus wavelength or fre-quency; these produce spectral comb fringes which createthe heterodyning effect through the Moire fringes. Thesemay or may not be resolved at the detector dependingon the disperser resolution, which is affected by its slitwidth. The invention does not need them to be resolved,but resolving them can improve signal to noise. 2) Versusposition along the slit. These are called “slit fringes” andexist when τ varies linearly along the slit, such as whenan internal interferometer mirror 25, or 29 or beamsplit-ter 19 is tilted. These fringes are not required but arehelpful. 3) Versus phase stepping. This is where τ is in-crement the same amount everywhere along the slit, suchas by moving mirror 29 by a PZT transducer 31. Theseare required to determine fringe phase and amplitude ifthere are no slit fringes. However, even when slit fringesexist phase stepping is useful for improving accuracy, es-pecially to distinguish common-mode errors.

2. Fixed Delay Size

For the embodiment of Fig. 1 that measures Dopplershift, the interferometer delay is approximately fixed(within a few waves), because that is simple, and suffi-cient to determine a Doppler effect by observing a phaseshift. This is in contrast with a conventional FourierTransform spectrometers which scan the delay over alarge range (tens of thousands of waves). The optimumfixed delay to use is approximately half the coherencelength (λ2/∆λ) of the spectral lines, where ∆λ is thetypical spectral linewidth of about 0.12 A. Linewidth isthe full width at half maximum (FWHM). This coherencelength is about 23 mm for starlight at λ=540 nm. Thischoice yields a good compromise between Moire fringevisibility and sensitivity of the Moire fringe phase to agiven Doppler velocity of target. (Larger delay will give

greater phase shift per velocity, but the fringe visibilitywill drop significantly when it is larger than this coher-ence length.) For use on sunlight and starlight, in oneembodiment, a delay of approximately 11.5 mm was used.

Note that this 23 mm coherence length due to the spec-tral lines is much larger than the approximately 1 microncoherence length of the continuum portion of the spec-trum. The inclusion of a disperser to an interferome-ter allows this larger coherence length component to besensed with reasonable fringe visibility.

Figure 3c shows that this delay choice is equivalent tomaking the half-period 500 of the comb fringes 505 aboutequal to the typical FWHM linewidth 502 of features inthe stellar spectrum 503. This way the “bright” half 504of the sine wave nicely overlaps the peak or dip 506 ofthe spectral line. The spectral comb period is given by

∆λ = (λ2/cτ). (2)

For a wavelength of 540 nm, a delay of 11.5 mm producesa comb periodicity of 0.25 A, which is indeed about twicethe typical stellar linewidth.

3. Wide-angle Ability from Superimposing Condition

In Fig. 1 the interferometer is a superimposing Michel-son design similar to that described by R.L.Hilliard andG.G. Shepherd, J. Opt. Soc. Amer., vol. 56, p362-369,(1966). The superimposing character improves the fringevisibility from extended sources such as wide fibers thatcan’t be perfectly collimated. Wide fibers are desired tocollect star images blurred by atmospheric turbulence.The superimposing type is where the mirror of one armand the image of the mirror of the other arm are su-perimposed longitudinally. This can be accomplished byinserting a transparent slab 21 (called the etalon) in oneof the arms. This creates a virtual image of the mirror25 which lies closer to the beamsplitter than the actualdistance. The mirror 29 of the other arm is arrangedto superimpose via the beamsplitter the virtual image ofmirror 25. This creates a non-zero temporal delay, whilesuperimposing the output paths of the rays from botharms. If n and d describe the refractive index and thick-ness respectively of the etalon, then the resulting delayat the superimposing condition is approximately

τc = 2d[(n − 1)/n + (n − 1)]. (3)

(The delay is a roundtrip length). For longer delays thanseveral centimeters, the etalon 21 can be replaced by areal imaging system.

4. Finding the Delay from the Data

The exact delay can be measured from the fringingspectra of a known spectral reference source, which couldbe an iodine vapor cell or a neon emission lamp. The

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wavelengths of the neon spectral lines are well known,and each delay value produces a particular combinationof phase shifts for the set of spectral lines. Alternatively,the broadband featureless lamp 3 is used for illuminationand the disperser’s slit is temporarily made very nar-row so that the spectral comb can be resolved. Thenthe number of comb fringes is counted from one end ofthe spectrum to the other. The spacing of the fringesis related to the delay. For any wavelength the absolutenumber of fringes that fit “inside” the delay is N = cτ/λ,so that counting the change in N across the spectrum isequated to cτ(1/λ1 − 1/λ2). This yields cτ .

5. Two Complementary Outputs Available

In general a Michelson interferometer produces twocomplementary outputs. In Fig. 1 only one is used, forsimplicity. The output traveling back towards the source(toward lens 13) is ignored. To improve efficiency how-ever, both outputs should be used. This can be done withschemes that distinguish the outputs by angles or polar-ization. If the ingoing beam enters at an angle to theoptic axis, then the outgoing beam will travel a differentpath than the ingoing and it is possible to introduce amirror to pick it off. An example scheme using polariza-tion is as follows: if the beamsplitter 19 is a polarizingbeamsplitter and the incident light is non-polarized orpolarized at 45 to the beamsplitter ( so that there issignificant intensity in both polarizations), and if meansto flip the polarization such as quarter-wave retardersare placed in each arm, then both complementary out-puts will travel along the same path toward lens 15. Theywon’t create fringes until they are separated by polariza-tion, which can be accomplished for example by a bire-fringent element downstream from the interferometer butprior to the CCD detector that causes one polarizationto strike a different place on the CCD than the otherpolarization.

6. Means for Phase-stepping the Delay

In general, a means for slightly varying the interferom-eter delay for all wavelength channels as a group shouldbe provided, so that the delay can be changed betweenexposures in evenly sized steps around the circle (360or 1 average wavelength). This is called phase-stepping.At least three exposures evenly spaced around the cir-cle are needed to uniquely determine fringe phase andamplitude for a given channel if the slit fringe period isinfinite. Steps of 90 however are sometimes more conve-nient than 120 steps. In this case, usually 4 or 5 expo-sures every 90 are made. Since the wavelength changesacross the spectrum, the actual detailed value of phasestep for a given delay increment ∆τ will vary with wave-length or frequency channel (∆φ = 2πc∆τ/λ = 2πf∆τ).The second reason for phase stepping is that it allows

common-mode errors that appear to be fringes (such asCCD pixel to pixel gain variations) to be distinguishedfrom true fringes, for only the true fringes will respondthe delay in a sinusoidal fashion versus stepped phase.

The means of stepping could be a piezoelectric trans-ducer (PZT) 31 which moves one of the interferometermirrors 29 or 25. For steps of 90 or less, the PZT couldbe linearly ramped during the exposure. This is sim-pler to implement than discrete stepping but washes outthe fringe amplitude slightly. For long exposures a sta-bilization scheme to step and then hold the phase stableagainst drifts is useful. The approximate phase of theinterferometer can be monitored by passing a laser suchas a HeNe laser through the same or similar portion ofthe interferometer cavity and recording the HeNe fringephase, (taking into account the wavelength ratio of theHeNe to the starlight.)

7. Temporarily Preventing Interference

A means for temporarily preventing interference orfringing behavior should be provided so that an ordinary(non-fringing) intensity spectrum can be recorded on theinstrument, while the keeping the rest of the instrumentas much as possible in the original configuration. This al-lows some common-mode behavior to be measured with-out the confusing effect of fringes, such as measuring thebeam profile at the slit and the CCD pixel to pixel gainvariation. These common-mode behaviors can then beused during data analysis to divide the fringing spectrumand thereby reduce the common mode errors. Otherwisebumps in the slit beam intensity profile due to vagariesin the optics that have similar periodicity to the fringescould be confused with the true fringes.

Interference can be prevented by blocking either of theinterferometer arms, arm A or arm B, such as by shut-ters 23 and 27 or opaque cards placed manually into thearms. Since the beam profile passing through arm A maybe slightly but significantly different than that passingthrough arm B, both nonfringing spectra (blocking A,blocking B) should be taken and then added later nu-merically.

8. Finite vs Infinitely Tall Spatial Fringes

When the two interferometer mirrors 25 and 29 arealigned so that the output rays from both arms super-impose, then an “infinitely tall” fringe is created on theoutput beam that entirely fills the height of the outputbeam measured along the slit length (Fig. 2a, item 56).When one of the mirrors is tilted from this condition, aparallel ladder 54 of fringes can be created. To observethe fringes by eye (without the disperser) a temporarymonochromatic source such as a laser or mercury lampcan be used. In Fig. 1, a lens system 15 images any lad-der of fringes effectively created at mirror planes 29 and

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Slitdirection

Subset fringingspectra

average = 0

average = 1

average = 2

Fig. 4 570

572

574

576

Figure 4 shows a fringing spectrum having many fringe peri-ods along the slit because the delay varies significantly.

25 to a plane near the slit 33. Figure 2b shows the ap-pearance to the eye of the fringe ladder 52 overlaying theslit 50. Figure 2a shows the intensity profile of the beampassing through the slit. The profile 54 has fringes andthe profile 56 is for the infinite fringe period case. Fig-ure 2c shows the action of the phase stepping is to cyclethe fringe ladder along the slit. Figure 2d shows that inthe case of infinitely tall fringes, the action of phase step-ping is to vary the average intensity sinusoidally versusstepped phase.

9. Best Size of Spatial Fringe

The optimum number of fringes in the ladder to spanthe beam is that which creates the most contrast rela-tive to noise effects. It depends on the imaging capa-bility (and astigmatism) of the disperser along the slitdirection. A poorly imaging disperser requires tall or in-finitely tall fringes. One should also avoid the spatialscales where the nonfringing intensity variations alongthe slit or across the CCD in the slit direction are large.In my apparatus measuring starlight I found having 4to 10 periods spanning the beam produces good contrastagainst noise. Since my beam occupies about 80 pixelsin the slit direction on the CCD, I have about 10 to 20pixels per slit fringe period.

10. Case of Very Many Spatial Fringes

In principle, if hundreds or thousands pixels of theCCD along the slit axis were used to record the slit in-tensity, then hundreds or thousands of fringes could berecorded along the slit. There is a logical distinction to bemade regarding the number of fringes in the slit directioncompared to the spectral resolution of the disperser. IfI have M fringes along the slit, the wavelength for thatchannel can be determined independent from knowingthe dispersion of the disperser by measuring the periodof slit fringes. This can be done to a fractional preci-sion of 1/M . That is, if 1000 fringes exist along the slit,then just from the fringes the relative wavelength of thatchannel can be determined to 1 part in 1000.

If this spectral resolution is finer than the spectral res-olution of the disperser (such as controlled by the slitwidth), and if the illumination spectrum contains fine de-tails of interest on this scale, then it is advantageous tosubdivide the fringing spectrum (570 in Fig. 4) vertically(the slit being defined vertical) into subset fringing spec-tra 572, 574 and 576 etc., each having a slightly differentaverage interferometer delay τ0, τ1, τ2 etc. Then thisis a method of creating a set of fringing spectrum takenwith a parallel set of different interferometer delays. Thismethod is similar to the method described later where astaircase-like etalon having different thickness segmentsis substituted for the single thickness etalon 21 in the in-terferometer cavity. Here, the parallel interferometers areeffectively created by the tilt of an interferometer mirror,because different portions of the CCD perpendicular tothe dispersion direction are sensing significantly differentinterferometer delays.

The motivation of using parallel delays is that by usinga wide range of delays the full information content ofthe illumination spectrum can be determined so that thespectrum can be mapped. A single delay having only afew fringes together with a low resolution disperser onlymeasures a subset of the full information. This subsetis sufficient to determine a Doppler dilation, but is notin general sufficient to map the spectrum at all levels ofdetail.

C. The Disperser

1. Combating Astigmatism

Some dispersers have intrinsic astigmatism which re-duces contrast of the fringes at the CCD detector in theslit direction. This astigmatism can be compensated forby focusing the fringes to a position slightly ahead orslightly behind the slit plane.

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2. Discussion

The disperser can be of any type, such as grating orprism. The necessary disperser spectral resolution is ap-proximately the typical feature to feature separation, toreduce crosstalk which mixes fringe phases and tends toreduce average fringe visibility. For stellar spectra thisis approximately 1 A. However, having a higher spectralresolution produces a better signal to noise if it can beobtained without sacrificing throughput, particularly if itis fine enough to resolve the spectral comb due to the in-terferometer by itself, about 0.25 A for a 11.5 mm delay.The optimum slit width thus also depends on the size ofthe source (fiber 5 diameter), since narrow sources allownarrower slit widths without reducing throughput.

3. Use of a 2nd Interferometer for Fiducials

It was mentioned that using a narrow slit to resolve thespectral comb improves the signal to noise of determiningthe whirl. This is because it creates a significant fringeamplitude in every wavelength channel, which helps de-termine overall common mode errors. However, in orderto efficiently collect light the slit should be wide. Thismay prevent resolving the spectral comb and hence theremay be channels with zero or little fringe amplitude. Thisproblem can be alleviated by artificially creating a newset of Moire fringes that are superimposed on the existingones.

This can be accomplished by including an additionalinterferometer into the beam path anywhere ahead of thedetector in Fig. 1, to act as an auxiliary spectral refer-ence. For example, it can be inserted ahead of the iodinecell 7 by using new appropriate imaging optics to reimagethe beam at point 11 to create a new small spot similarto point 11. The delay of the additional interferometercan be chosen to be slightly different than the delay ofthe original interferometer. The sinusoidal spectrum ofthe new interferometer multiplies the existing spectrumof the beam and creates new Moire fringes and a newcomponent to the net whirl. These new Moire fringes areeasily distinguished from the random-like Moire fringesdue to the target spectrum by their regular sinusoidalnature. Thus, the detailed value of the new delay doesnot have to be known. A secondary benefit is that it al-lows some instrumental error to be determined. Since thenew Moire fringes should theoretically be perfectly regu-lar, any deviation of the measured fringe can be assumedto be due to instrumental artifacts.

As an example, if the instrument bandwidth is 6.5%(350 A out of 5400 A), then 16 waves of delay changecreates 16 × 6.5% = 1 wave of Moire fringe across the350 A spectrum. Thus a delay difference between the newand original interferometers of 1600 waves will create aMoire pattern with 100 waves spanning the spectrum

IV. THEORY OF OPERATION

A. Moire or Heterodyning Effect

The invention takes advantage of the Moire effect be-tween the sinusoidal frequency behavior of the inter-ferometer and any similar-period sinusoidal-like compo-nents of the illumination spectrum. This occurs becausethe sinusoidal transmission function of the interferom-eter multiplies the illumination intensity spectrum cre-ating sum and difference “frequency” components, andthe sum-frequency components are eliminated by eitherthe blurring of the disperser or equivalent blurring donenumerically during data analysis. (Warning, the use ofthe term “frequency” in this paragraph is different thanelsewhere in the document because it’s a frequency of afunction that is already in frequency-space.) The Moireeffect heterodynes high spectral detail to low spectral de-tail, allowing the use of a low resolution disperser to ef-fectively detect high resolution phenomena. The Moirefringe phase and amplitude versus wavelength channel isrepresented by a set of vectors called a “whirl”. Anotherway to describe the heterodyning is to take the Fouriertransform of the illumination intensity spectrum to pro-duce an interferogram. The Moire effect translates theinterferogram horizontally by a constant offset τk whichis the fixed interferometer delay used.

If the goal is to map a spectrum, then during dataanalysis the heterodyning process can be reversed math-ematically, beating the whirl up to higher “frequencies”to restore the original high detail information. If thegoal is to measure a Doppler dilation of the spectrum fora fixed delay, or equivalently a change in delay for a fixedspectrum, then the up-heterodyning is unnecessary. Thedilation or delay change is determined from the rotationof the Moire whirl.

B. Interferometer Spectral Comb

Figure 3a and 3b show the 2-dimensional spectrumof white light after passing through an interferometerof non-zero delay, frequency versus position along theslit. Pretend for the moment that the disperser has per-fect spectral resolution– no blurring. A spectral fringecomb 67 exist along the dispersion direction having pe-riod ∆f = 1/τ , which for an 11.5 mm delay correspondsto a period of 0.25 A of green light (5400 A). In the caseof Fig. 3b the interferometer mirrors are in alignment sothat there is a tall fringe 66 completely covering the slit.This makes a comb 64 which has vertical fringes parallelto the slit. Figure 3a shows the case when an interferom-eter mirror is tilted so that several fringes 65 are createdalong the slit. This creates a slanted comb 63.

Figure 5a shows how overlapping the slanted fringecomb 70 with the vertical features 72 of a stellar spec-trum creates Moire fringes. These appear as beads alongthe slit direction if one squints ones eyes at the Figure. A

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Slit

dire

ctio

n

b)a)

Fig. 5

Slit

dire

ctio

n

Slit

dire

ctio

n

c)

72

70

Figure 5a shows through 5c show the generation of Moirefringes from a slanted fringe comb overlaying spectral lines.

Slit

d)

Figure 5d shows a whirl of Arturus starlight data displayedas pseudofringes.

Doppler effect will shift the features slightly in the wave-length direction. This causes the phase of all the Moirefringes to shift vertically, almost uniformly as a block.Not exactly as a block because the phase shift is propor-tional to frequency, so it grows linearly from the red toblue ends of the spectrum. In my apparatus for exam-ple, the spectrum spans 350 A centered about 5400 A,so the Doppler phase shift on one end of the spectrum is7% different than the other end. When the Moire fringedata are expressed as a whirl, a Doppler shift causes thewhirl to rotate with a little bit of twist.

Figure 5b would be the appearance when the disperserresolution is made coarser to blur away the spectral comb.Figure 5c would be under further blurring, which couldbe done numerically when the data is processed.

C. Benefit of Randomness of Systematic Errors

Figure 6a shows the benefit of illumination such asstarlight or the back-lit iodine spectrum which have spec-tral features whose detailed wavelengths are randomlyrelated, and the interferometer delay is sufficiently largeto cause the phase of these fringes to be randomly andevenly distributed around a cycle. The advantage is

a)

Slit

dire

ctio

n

b)

Fig. 6Positive error

Negative error

104

106

100

102

Figure 6a and 6b show the vector angles and fringing spec-trum when the fringe phases are randomly distributed over alarge range.

Slit

dire

ctio

n

d)c)108

109

Figure 6c and 6d show the vector angles and fringing spectrumwhen the fringe phases are too similar.

that some systematic errors are reduced because theyare evenly sampled over all phase-space. Figure 6b showsa hypothetical fringing spectrum with randomly relatedphases. Figure 6a shows that the phases 106 of thosefringes are randomly distributed about a circle. Oftenin interferometry, a systematic error will exist which isdependent on phase. For example, if two signals whichare supposed to be in quadrature (so that the arctangentfunction can be applied to measure their phase angle) arenot perfectly in quadrature, then an ellipticity error willresult, which has polarity which is positive 100 for somephase angles, and negative 102 for others. Having manyfringes sampling all kinds of angles will reduce the netsystematic error by a factor of square root of the numberof independent fringes, statistically. This can be a signif-icant reduction when many hundreds of stellar or iodinelines are used.

In contrast, Fig. 6d and 6c show fringe phases 108which are too similar to each other, such as when a smalldelay is used, or if the illumination is too monochromatic,or if the continuum portion of a stellar spectrum is usedinstead of the narrow spectral lines (which is the casefor conventional long-baseline interferometers). The sim-ilarity of the phases causes the systematic error to besampled too coarsely, predominantly on one segment 109of a cycle, increasing the net systematic error.

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a) "Whirl"

Y

X

R

b)

120

Fig. 7

122Y

X

Figure 7a shows the whirl as a meandering path traced by awavelength dependent vector. Figure 7b shows the relationbetween polar and rectangular coordinates.

D. Relation Between Velocity and Moire Rotation

The relation between the target velocity and whirl ro-tation is as follows. For a target moving with velocityv toward the apparatus, the wavelength spectrum ap-pears contracted (more energy in the blue) by a factor(1 + v/c). This is equivalent to having the delay appearlarger by that same factor. This an increase in delay by∆τ = τ(v/c). This is a phase shift (which is equivalentto rotation of a vector in radians) by

∆φ = 2πc∆τ/λ = 2πτv/λ. (4)

The velocity per fringe proportionality is (λ/τ) = 14090m/s for a 11.5 mm delay and 540 nm light.

V. DATA TAKING PROCEDURE

To measure the Doppler velocity of the sun or star onemust measure three kinds of sources 1) the sun/star byitself without the iodine cell, to produce a “sol” whirl;2) the iodine by itself without the sun/star, illuminatedby a featureless lamp, to produce an “io” whirl; and 3)the sun/star with the iodine cell to produce a “solio”whirl. The order of these three steps is unimportant.The terminology comes from the use of sunlight as a testsource that has a similar spectrum as starlight. The soland io whirls need only be measured once, and they aretreated as reference whirls. For each solio whirl produced,a Doppler velocity can be determined, using the same soland io reference whirls.

For each source, ideally four or five fringing spectra ex-posures are recorded with phase steps of approximately90. Also, for each source a nonfringing spectrum isrecorded by blocking arm A, and another by blockingarm B.

VI. MAKING THE WHIRL FROM THEFRINGING SPECTRUM

A. Finite Fringe Period Case

For the case of finite slit fringe periodicity, the whirlis obtained from the fringing spectrum by evaluating foreach wavelength channel the Fourier sine and cosine am-plitudes of the intensity profile along the slit direction atthe expected fringe periodicity. This is easily done bymultiplying each vertical column of CCD data by a sinewave or cosine wave having the expected periodicity andthen summing along the column. To compute Fourieramplitudes properly, the boundary conditions should beperiodic. Hence, one should limit the number of pixels inthe column to be a integer number of periods. Prior tocomputing the Fourier amplitude, it is beneficial to nor-malize the data by dividing out the average intensity pro-file along the slit. The resulting fluctuations around theiraverage are multiplied by a smooth enveloping function.This envelope smoothly goes to zero at each boundarypoint, to minimize sharp discontinuities.

The Fourier sine and cosine amplitudes are assignedto be the rectangular coordinates X, Y of a vector (120in Fig. 7), for each wavelength channel. In polar coor-dinates, the vector length (R) and angle (θ) representthe fringe amplitude and phase, respectively. The set ofvectors over all channels is called the whirl. Figure 7ashows the whirl as a meandering path 122 traced out bya wavelength dependent vector, and Fig. 7b the relationbetween polar and rectangular coordinates. Figure 5dshows a measured whirl of Arcturus starlight over ap-proximately 140 A near 5400 A. This shows only 1000channels of the 2500 channel CCD. In this method ofdisplay, the phase and amplitude are represented by ar-tificial fringes in a calculated intensity map.

B. Infinite Fringe Period Case

For the case of infinite period slit fringes, standardphase stepping algorithms may be used to determinefringe phase and angle for each wavelength channel, us-ing several exposures with incremented delays distributedaround the circle. For example, suppose I1, I2, I3, andI4 are the intensities for a given λ channel taken every90 in phase, sequentially, where the intensity is summedor averaged over all the pixels of a channel. Then thetangent of the average phase angle is given by

tanφ = (I2 − I4)/(I1 − I3), (5)

and the average amplitude R given by

R2 = (I1 − I3)2 + (I2 − I4)2. (6)

If five exposures are taken, then I1 can be replaced by theaverage of I1 and I5. This has the advantage of makingφ less sensitive to changes in the phase step amount due

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io

sol1

∆sol

solio1

solio2

sol2200

202204

206

208

210

Fig. 8

∆ D

Figure 8 shows the vector relationships of the sol and is whirlcomponents under effect of a Doppler rotation.

to twisting versus wavelength. Note, the ideal 90 step isan average value– the actual step value is proportional tothe frequency (c/λ) at the channel, which changes acrossthe spectrum.

C. Phase Stepping to Reduce Fixed-pattern Errors

Now back to the case of finite fringe period. Eventhough phase stepping is not required to determinedfringe phase and amplitude, phase stepping is recom-mended as an additional step because it can be used tocompute an average whirl which has reduced common-mode errors. These are errors that do not move syn-chronously with the phase stepping, such as effects ofbumps in the slit intensity profile due to beam aber-rations, and that escaped complete removal during thenormalization step. Let W1 through W4 be the indi-vidual whirls computed as described above via Fourieramplitudes, one whirl for each phase step. First one ro-tates whirls W2 through W4 backwards by the same phasestepped angle each was exposed at, so that all the whirlshave the same approximate rotation as W1. Implement-ing this rotation is easiest if the whirl vectors are in po-lar coordinates, by offsetting the angle parameter. Thenone converts each vector from polar to rectangular coor-dinates. Then one simply computes the average whirl byvector-adding all the whirls together while in rectangularcoordinates and dividing by the number of whirls. If theexposures are taken at even positions around the circle,such as 3 exposures every 120 or 4 exposures every 90,then the common-mode error component will be zeroedor greatly reduced by this rotate-then-sum process.

Also, as above, a fifth exposure can be taken at the360 phase step position, and averaged together with the0 exposure before substituting for the original 0 expo-sure in the whirl averaging process. This helps reduce theaffect of a wavelength dependent phase step by increasingthe symmetry of the problem.

D. Finding Whirl Rotation for DeterminingDoppler Shift

The data analysis procedure for determine the Dopplershift will be described. The first step is to prepare thewhirls so that they are approximately aligned, to removeany coarse effects of various drifts. The alignment oc-curs in rotation, twisting, radial magnitude and horizon-tal (wavelength) translation. These drifts have nothingto do with Doppler shifts and therefore can legitimatelybe removed. Because the target is measured simultane-ous with the iodine spectrum, ideally the drifts shouldeffect both whirl components, sol and io. However, inreality because the iodine lines and stellar lines don’t oc-cur at exactly the same place, the drifts can slightly affectthe Doppler result, and therefore it is best to remove thedrifts when possible.

During this alignment procedure, a solio whirl, usuallythe first, is designated to be the “standard” to whichall other whirls are aligned. Then the input whirl tobe aligned is rotated, twisted, expanded in magnitudeand translated horizontally to essentially minimize theroot means square difference between it and the standardwhirl. This is only an approximate process meant toremove gross drifts. all the solio whirls and the sol andio reference whirls are aligned this way to the standardsolio.

Figure 8 illustrates that a Doppler effect rotates the solwhirl component relative to the io whirl component by anangle ∆φ. The vectors of Fig. 8 represent whirls, whichare themselves aggregates of vectors. It is presumed thatthe solio whirl is a linear combination of sol and io whirls,added vectorally channel by channel. Suppose solio1 andsolio2 are the solio whirls measured on two different oc-casions. Figure 8 supposes that these whirls have alreadybeen aligned so that their io components 210 are super-imposed. Then a Doppler effect rotates the sol2 compo-nent (202) relative to the sol1 component 200. Therefore,the difference between solio1 and solio2 is a small vector∆sol (208) which is approximately perpendicular to sol1or sol2. The phase angle due to the Doppler effect, ∆φ,can be found by taking the arctangent of length of ∆solrelative to the length of sol1. The length of ∆sol can befound by applying dot products, since it is known that∆sol is perpendicular to sol1.

1. Dot Product Between Two Whirls

The dot product between two whirls is defined tobe the individual dot product of the two vectors for achannel, summed or averaged over all channels. Let awhirl in rectangular coordinates be the set of vectorsW = [X(λ), Y (λ)], where X, Y are the cosine and sinefringe amplitudes respectively. Then the dot product be-tween two whirls W1 and W2 is

W1 · W2 =∑

X1(λ)X2(λ) + Y1(λ)Y2(λ) (7)

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summed over all λ channels. This could be called a “gen-eralized” dot product because it sums products over boththe spatial and wavelength indices.

2. A More Exact Dot-product Analysis

The procedure of Fig. 8 was a simple approximationthat made assumptions about the constancy of the mag-nitude of sol and io components. A more exact analysisis now described. This solves for the rotational posi-tion of the sol whirl component relative to the io whirlby application of (generalized) dot products and solvingthe resulting 4 equations in 4 unknowns. The input tothe process is a solio whirl and the output is an angu-lar difference which can be converted to a velocity. It isassumed that the sol and io whirls have previously beenmeasured and are used as reference whirls, the same eachinput solio.

The solio whirl is assumed to be a linear combinationof 4 components: sol, sol⊥, io and io⊥, where sol⊥ andio⊥ are perpendicular whirls made by rotating sol and ioby 90. (This can be conveniently done by switching theX(λ) and Y (λ) components and flipping the polarity ofone of them.) If low pass filtering is not performed and asignificant spectral comb component due to the interfer-ometer on the continuum is present, then two additionalterms of comb and comb⊥ can be added. The analysisbelow assumes these terms are not needed because thesespectral comb components have been filtered away.

Let a solio be expressed as the linear combination:

solio = Asio + Atio⊥ + Bssol + Btsol⊥, (8)

where the coefficients As, At, Bs, and Bt hold the mag-nitude and rotational information. For each solio whirlmeasured the following set of dot products are computedusing the same io and sol whirls, measured separately:

Tik = io · solio (9)Tipk = io⊥ · solio (10)Tsk = sol · solio (11)

Tspk = sol⊥ · solio (12)

Hence a set of Tik etc. coefficients are computed for eachsolio to be processed.

Previously, the dot products involving the referencewhirls have been computed:

T1 = io · sol (13)T2 = io · sol⊥ (14)T3 = sol · sol (15)T4 = io · io (16)

Now we have 4 equations in 4 unknowns (As, At, Bs,Bt) which can be exactly solved by standard linear al-gebra procedures, such as by “Kramers rule” which uses

determinants. The 4 equations are:

Tik = T4As + 0 + T1Bs + T2Bt (17)Tipk = 0 + T4At − T2Bs + T1Bt (18)Tsk = T1As − T2At + T3Bs + 0 (19)

Tspk = T2As + T1At + 0 + T3Bt (20)

Once (As, At, Bs, Bt) have been determined for agiven solion, then the values for the angles φio and φsol

are found corresponding to that solio, which describe therotational orientation of the component io and sol withinsolio. The values are found through

φio = arctan(At/As) (21)φsol = arctan(Bt/Bs) (22)

The Doppler shift ∆φD involves the rotation of solrelative to io. Hence,

φD = φsol − φio (23)

is the net angle between sol and io; and the change inthe net angle between two occasions 1 and 2 is

∆φD = φD2 − φD1 (24)

The Doppler velocity ∆v is found from the angle ∆φD

by

∆v = (∆φD/2π)c(λ/cτ), (25)

where λ is the average wavelength.

3. High Precision Verified by Experiment

The capabilities of this apparatus have been testedby measuring Doppler velocities of a stationary sourceconsisting of a bromine lamp back-illuminated by whitelight. Over 20 minutes and seventeen independent mea-surements the repeatability of the measurements was0.76 m/s, of which a significant portion may have beenphoton noise. This corresponds to a broadband phaseshift repeatability of 1/20,000th of a wave, which is muchbetter than conventional phase shift measurements usingmonochromatic illumination.

E. Metrology Based on Dilation of anInterferometer Cavity

Figure 9 shows an embodiment of the invention formeasuring secondary effects such as temperature, airpressure and acceleration which can be made to changean interferometer delay, such as through motion of a cav-ity mirror or the refractive index of a medium internal tothe interferometer. This operates on the principle that ifthe illumination spectrum is constant, then a rotation ofthe whirl must be due to the delay change

c∆τ = (∆φ/2π). (26)

6-16

244

246

I2 vapor cell

232

Perspective view ofdisperser

Fringing spectra

230

Sensing interferometer

238238

240

242

236

Fig. 9

Fiber bundle

Dilative effect to be sensed: heat,magnetic field, acceleration, airdensity etc.

/2

white lightsource

231

Quadrature phasesensing

Fringes

233234

237

239

235

Figure 9 shows an embodiment that measures a secondary effect through changes in interferometer delay using broadbandillumination.

A convenient spectrum to use is the iodine absorptionspectrum illuminated by featureless light. This is pro-vided by the iodine cell 234 back-illuminated by a whitelamp 230. The iodine cell can be anywhere along theoptical path. It may be convenient to have it just aheadof the disperser, rather than near the interferometer, be-cause presumably it is desirable to make optics near theinterferometer compact so that it can be used as a remoteprobe at the end of an optical fiber 235. Figure 9 uses alow-finesse Fabry-Perot interferometer as the interferom-eter to illustrate that it is possible to use non-Michelsoninterferometers to produce sinusoidal-like fringes pro-vided means are employed to discriminate against thehigher harmonics of the fringe shape. Such means in-clude phase stepping at 90 steps or sampling the slitfringe at quarter wave intervals. Fabry-Perot interferom-eters are advantageous because they are compact. The fi-nesse must be low, which is accomplished by using poorlyreflective mirrors in the Fabry-Perot cavity, for example,a reflectivity of 27%.

In Fig. 9 the light from lamp 230 passing through apinhole 231 creates ring-like fringes 237 due to the inter-ference of multiple reflections 240, 242 between the twopartially reflecting mirrors 238 that define the interferom-eter cavity. The delay is equal to the roundtrip length

of a ray between the mirrors. The effect to be measuredis presumed to alter the optical path length between thetwo mirrors. For example, temperature could dilate thelength of the spacer 236 that controls the mirror spacing.A means for sampling the ring fringes every quarter waveat 239 (quadrature sensing) is provided by the 4-fiberbundle 235. A lens 233 images the four fiber outputsto separate spots 232 on the disperser slit. A detectorwith an imaging disperser symbolically represented bythe prism 244 records a fringing spectrum for each of thefour quadrature sensing spots 238.

Figure 10a through 10c show how the non-sinusoidalcomponent of the Fabry-Perot fringes can be discrimi-nated against using quadrature sampling along the “slit”.Figure 10a shows the fringe intensity across the diameterof the ring pattern for a Fabry-Perot with mirror reflec-tivity approximately 27%. It has a large fundamental si-nusoidal component (Fig. 10b), but with significant com-ponents at higher harmonics, such as the 2nd harmonicshown in Fig. 10c. The principle is to sample the fringewith few enough samples so that only the fundamentalperiod is resolved, and the shorter spatial period har-monics are not resolved because they do not satisfy theNyquist criteria. When the intensity is measured every90, then the fundamental is resolved, but the 2nd and

6-17

I1

I2

I3

I4

I1I2

I3I4

I1

I2 I3 I4

Across fringes

Inte

nsity

0

Fabry-Perot fringesa)

b)

c)

Fundamental

2nd harmonic

Reflectivity=0.27

Fig. 10

Figure 10a through 10c shows how to use the sinusoidal com-ponent of a low-finesse Fabry-Perot interferometer.

d) PZT spacer

Three interferometer cavities

Three superimposedfringe patterns

A B C

HousingSensing spacer

Fiber or fibers

271 272 270273

Figure 10d shows the inclusion of an additional cavity as ameans for phase stepping.

higher harmonics have only 2 samples per period, and soare unresolved. For example for the second harmonic, ifwe apply the R2 = (I1 − I3)2 + (I2 − I4)2 relation to findthe fringe amplitude, Fig. 10c shows that the differences(I1 − I3) and (I2 − I4) are zero.

Another method of quadrature sampling is to phasestep every quarter wave. This can be done with a sin-gle fiber instead of a 4-fiber bundle. However, becausethe delay is now being changed, an independent methodof determining the delay accurately is needed. This canbe provided by inserting an additional partially reflectivesurface C in series with the first interferometer, and havethe secondary effect affect only one of the interferometercavities AB. In Fig. 10d, a partially reflective surface 270(labeled “C”) is included in addition to the existing sur-faces A (271) and B (272). A PZT transducer 273 movesthis surface to create phase stepping. Three Fabry-Perotcavities exist instead of one: A to B, B to C, and A to C.Three sets of ring fringes will be produced. Thus the netwhirl derived from this apparatus will have three com-ponent whirls corresponding to the three interferometerdelays, τAB , τBC and τAC . The delays are chosen to be

different enough so that the whirls are approximately or-thogonal. Orthogonal means that their dot product isseveral times smaller than either self-dot product. Thisoccurs when the delay is many waves different from eachother so that there is at least one cycle of twist. For ex-ample, if the spectrum spans a 5% change in wavelength,then 20 waves (average wavelength) of delay differencewill create 1 cycle of twist, which makes the two whirlsorthogonal. Greater number of twists are preferred toimprove orthogonality.

A dot-product analysis analogous to that describedabove for the Doppler effect can be used to determinethe rotations of the three whirl components correspond-ing to the three delays. The twists can be anticipated bytwisting the reference whirls by an appropriate amount.Then the delay change corresponding to cavity AB yieldsthe secondary effect to be measured.

VII. MEASUREMENT OF ANGLES USING ALONG-BASELINE INTERFEROMETER

Long baseline interferometers use phase shifts to mea-sure angular positions of a distant targets. Hence, an em-bodiment of this invention substituting a long baseline in-terferometer for the interferometer can measure angularpositions very accurately, by associating whirl rotationand twist with target angle. Figure 11 shows an embod-iment that incorporates a long baseline interferometer.The interferometer consists of the triangle formed by thetarget 300, and the left 302 and right 304 collection ports.(In order to draw the nearly parallel rays of light fromthe distant target, the target 300 is drawn twice.) Thecollection ports are spaced a distance D apart, called thebaseline distance. The light entering the collection portsare conducted via paths 314 and 316 to a beamsplitter306 where the light interferes with light from the othercollection port.

Since the two streams of light originated from the samestar at small angle, they are nearly identical but arriveat different times depending on the angular position θof the target relative to the baseline. Suppose for themoment that the two optical paths 314 and 316 havethe same length, so we can ignore their contribution tothe arrival time difference. The arrival time differenceforms the delay τ of the invention, and the inclusion ofa disperser 308 creates a fringing spectrum recorded ata CCD detector 310. Tilting the beams slightly withrespect to each other at the beamsplitter creates fringesacross the slit of the disperser. A mirror moved by a PZTtransducer that changes the path of one of the beamsprior to interfering could provide the phase stepping.

1. Iodine Cell Imprints a Reference Spectrum

Narrow spectral features are desired to measure largeangles. Although it is possible to use the intrinsic spec-

6-18

I2

B

TargetSpectrometer

CCD

Beamsplitter

∆

Target

Figure 11a)

Fringing spectrumrecorded

The same object

300

300

304314

302

316

306

308

310

312

Figure 11a shows an embodiment that measures angles through phase shifts of a long baseline interferometer.

tral lines of the star (target), if the star is moving theDoppler effect may confuse the angular determination.Hence it is much better to insert an iodine vapor cell312 into the beam to imprint a known and stable spec-trum. This allows measuring angles of targets lackingspectral features, not limited to stars. Furthermore, thelinewidths of iodine are about 8 times narrower than stel-lar lines, so that the maximum delays can be 8 timeslonger (about 80 mm). This allows measuring angles 8time larger than without the iodine cell. (The stellarcomponent of the whirl can be ignored because it is suffi-ciently orthogonal to the iodine whirl for large bandwidthspectra. Hence, lets omit further discussion of the stellarcomponent.)

2. Relation Between Target Angle and Whirl Rotation

Let an earlier measurement of the target be definedas a reference whirl. Then the rotational position ofthe current target whirl relative to the reference canbe determined by taking dot products against the ref-erence whirl an its perpendicular whirl, analogous to theDoppler analysis described earlier. The reference whirlmay have to be artificially twisted in order improve thealignment with the target whirl if the angle is large. Theresulting whirl angle change ∆φ is related to the targetangle change by

∆θ = (λ/D)(∆φ/2π) (27)

Alternatively, a reference whirl is constructed mathe-matically from the known iodine spectrum. This can bedone by Fourier transforming the iodine spectrum to forman interferogram, translating it by a delay τ0, mimicingthe enveloping action of the disperser blur, and then in-verse Fourier transforming to form the theoretical whirl.The τ0 that produces the best agreement with the mea-sured whirl yields the angle through

θ = cτ0/D, (28)

if the paths 314, 316 are of equal length. The rotationalpolarity of the whirl (which is controlled in an actualinstrument by the polarity of mirror tilt) will determinethe polarity of the angle θ.

3. Works with Non-zero Angles

Note that the technique works with arrival times thatare significantly non-zero, and cannot work with exactlyzero arrival times because then the spectral comb is toobroad compared to the spectrum’s total width (the con-tinuum portion). This is distinct from conventional longbaseline interferometers which work best for near zeroarrival times.

4. Use of Reference Stars to Take Angular DifferenceIndependent of Pathlength Drifts

The paths 314 and 316 may be different by someamount ∆s, which directly adds to the apparent delayτ . In practice it may be difficult to measure or stabilize∆s to sufficient accuracy. Furthermore, when slit fringesor phase stepping is used this creates an uncertainty inthe delay. Figure 11b shows that a solution is to use asecond star (referenceA star, 318) and have light of bothstars travel through the same paths 314 and 316. Thiscan be accomplished by using beamsplitting systems tocombine light at the collection ports 302 and 304. Thenet whirl will then contain two component whirls, whichcan be distinguished during the dot product procedurewhen the angles are sufficiently different, so that thereare at least one twist in the whirl relative to each other.In other words, so that whirls are sufficiently orthogo-nal that they can be distinguished from each other. Thismethod yields a difference in angles between the targetstar 320 and referenceA star 318. The slit fringes canbe implemented by tilting the angle at which the light

6-19

I2

D

AReferenceA

ReferenceC

TargetBSpectrometer

CCD

Light combined atentrance of opticalpath

BeamsplitterC B

Fringing spectrumwith component whirls

318

320

322

318

320

322

b)

Figure 11b shows another embodiment that measures angles through phase shifts of a long baseline interferometer.

from the two paths 314 and 316 join each other at thebeamsplitter.

In a similar way, the baseline distance D may be un-certain. This uncertainty can be removed from the prob-lem by observing a third star, and having its light passthrough the same paths as the other two stars, by meansof beamsplitting at the collection ports 304 and 306. Thiscan measure the angular distance of the target star 320relative to the angular separation between the two refer-ence stars 318 and 322.

5. Distinction from Prior Art

The technique is distinct from the spectroscopicmethod of Kandpal in several important ways:

1. The Kandpal technique does heterodyne or produceMoire fringes between the interferometer and anynarrow spectral features of the target spectrum.The modulations created by the interference areresolved directly by the disperser.

2. An iodine cell or other imprinted reference spec-trum having narrow lines is not used in the Kand-pal method.

3. There is no phase stepping or slit fringes. Conse-quently, fringe phase and amplitude (2-d vector in-formation) is not obtained from a given λ-channelindependent of others. Only a scalar (intensity)spectrum is measured. The modulations they ob-served are versus wavelength channel rather thanversus delay. Hence, the phase and amplitude of afringe for a given wavelength channel in isolationfrom other, cannot be determined without priorknowledge of the intensity spectrum. This is espe-cially critical when the intrinsic (nonfringing) stel-lar spectrum has narrow dips and peaks of similarwidth as the sinusoidal modulations, such as is thecase when larger delays are used.

Because only scalar and not vector information ismeasured for each wavelength channel, the polar-ity of the modulations cannot be determined, sowhether light from a star comes from the left orright of the perpendicular to the baseline cannotbe distinguished.

4. The interferometer delay for Kandpal is limited tovery small values of about 10 microns, set by the re-ciprocal of the spectral resolution of the disperser.In contrast, in my invention the delay can be 8000times larger (80 mm versus 10 microns) becausethe iodine spectrum is heterodyning against the in-terferometer spectral comb. This means 8000 timeslarger angles can be measured for the same baseline.This is a significant practical advantage because itmakes it much more likely to find neighboring starsto use as positional references.

VIII. SPECTRAL MAPPING

A. Stepped Mirror and Etalon

An embodiment of the invention which is useful formapping a spectrum is shown in Fig. 12a. The single de-lay is replaced by a parallel set of delays having a rangeof values that cover the coherence length of the illumina-tion. An internal interferometer mirror 25 is replaced bya stepped or staircase-like mirror 362 which is segmentedinto steps of different thickness labeled A through E. Fur-thermore, in order to preserve the superimposing condi-tion which improves fringe visibility, the etalon 21 of theinstrument in Fig. 1 is optimally replaced by a staircaseetalon 360 which is segmented into steps also labeled Athrough E that correspond to the steps of the staircasemirror. Each segment A of the mirror 362 and etalon 360is imaged to an independent location of the disperser slit370 by a lens system 364. Similarly, a lens system 366

6-20

M1

SourcePlane(Slit-like)

ABCDE

ABCDE

ABC

ED

ApparentMiirror plane

Input slit ofspectrometer Spectromete

r Dispersiondirection intothe paper

CCD

BS

L1

L2Staircaseetalon

Staircasemirror

PZTTranslator

a) Fig. 12368

366

364

370

374

372

360362

376

Figure 12a shows an embodiment that creates multiple simultaneous fringing spectra having different delays.

Sr

Se

A

B

C

D

E

ApparentMiirror plane

Staircaseetalon

Staircasemirror

b)

376

Se / Sr ratio determinedby glass refractive index.Choosing this ratio keepsApparent Mirror Planeposition constant for allsteps.

Figure 12b shows another embodiment that creates multiplesimultaneous fringing spectra having different delays.

images a slit-like source 368 to the segments, presentingthe same spectrum to all segments.

The stepped nature of the etalon and mirror are not tobe confused with a grating, because the beamlets fromthe etalon segments do not interfere with each other, asthey would in a grating. Instead, they are imaged toseparate locations on the CCD to form separate fringingspectra. All these fringing spectra can be phase steppedsimultaneously by the PZT 372, and have the same slitfringe periodicity by tilting mirror M1 (374).

Figure 12b shows that the individual etalon segmentthickness Se and mirror segment location Sr are chosenso that for each segment the apparent mirror segmentso-created lies in the same plane 376. This is done by

a)

0

delay-space

∆IntensityIntrinsic interferogram of illumination

Coherence length

Figure 13

b)

0

Portion measured by low resolution disperser alone

delay-space

∆T

406

404

403

402

400

Figure 13a through 13d shows how to reconstruct an interfer-ogram from interferogram segments made from whirls.

setting

Se(n − 1)/n = Sr (29)

(The apparent mirror is the actual mirror shifted forwardby the virtual imaging of the etalon slab.) This apparentmirror 376 is optimally superimposed with the mirrorM1 (374) of the other interferometer arm by action ofthe beamsplitter.

6-21

c)

0

Portion measured by disperser with interferometer of delay k

delay-space

∆T

= k

d)

0

Reconstructing the interferogram from concatenated segments

delay-space

= 1

= 2

408

410411

412 413 414 415 416

Figure 13d shows the concatenating step in reconstructing aninterferogram from interferogram segments made from whirls.

B. Explanation of Behavior through theInterferogram

Figure 13a represents an interferogram 400 of the in-strinsic illumination spectrum at the slit 368, with per-fect spectral resolution. An interferogram is the Fouriertransform of the intensity spectrum. It’s shape containsall the information needed to reconstruct a spectrum.(It will be symmetrical about zero delay, so only one sideneeds to be shown or measured. This is consequence ofthe intensity spectrum being a real function.) An in-terferogram is useful to explain the action of the het-erodyning process that creates a whirl, and the spectralresolution of a system.

Figure 13b shows that a disperser that has low spectralresolution only preserves the portion 402 of the originalinterferogram 400 in a small range 404 near zero, havingwidth ∆T . The goal of a high resolution spectrometersystem is to preserve all the interferogram informationby having a large range, which is at least as large asthe illumination coherence length 406. Figure 13c showsthat due to the heterodyning process that creates Moirefringes, the Fourier transform of a whirl created withdelay τk will be the portion 408 of interferogram 400,centered at τ = τk with width ∆T . Hence, Fig. 13d showsthat with a set of parallel delays of different values τa,τb, τc, τd etc., the apparatus of Fig. 12a can measure thewhole range of the interferogram using a contiguous setof segments (410 through 416), the delay values chosen sothat each range is shoulder to shoulder with its neighbor.

C. Data Analysis for Reconstructing the Spectrum

Hence the data analysis procedure for mapping a spec-trum is as follows: For each whirl Wk measured with de-lay τk, the Fourier transform is performed to create an in-terferogram segment Qk. This segment is shifted toward

increasing τ by the amount τk to form an adjusted inter-ferogram segment. After this is done for all the whirls,all the adjusted segments are concatenated together toform a 1-sided concatenated interferogram. The 2-sidedinterferogram must be symmetric about zero delay, sincethe intensity spectrum is known to be a real function.Therefore the next step is to copy the 1-side to the otherside by reflecting about zero delay to form the full spec-trum. This is then inverse Fourier transformed to formthe net output whirl. The magnitude (vector length) ofthis whirl versus wavelength channel forms the outputintensity spectrum.

The Fourier transforms of whirls should be a com-plex Fourier transform. Since frequency and delay τare Fourier transform pairs, I should use here the ter-minology of “frequency” instead of “wavelength” to bemathematically correct. Let each frequency (wavelength)channel be designated by an index f . The vector of eachf -channel needs to be expressed in rectangular coordi-nates, so that the whirl consists of real and imaginaryfunctions of f having N points, where N should be evenand ideally padded by zeros so that it is a power of two tospeed computation. The discrete fast Fourier transform(FFT) of a complex function W (f) having N points pro-duces another complex function Q(τ) of N points. FFTalgorithms are well known. For example,

Q(τ) =∑

W (f)ei2πfτ/N , (30)

where the sum is over f = 0 to f = N − 1.The concatenation process needs to use appropriate

enveloping to undo the non-constant enveloping effect ofthe disperser, so that the transition from one adjusted in-terferogram segment to the neighboring one, which mayoverlap, is smooth and does not artificially emphasizeor de-emphasize portions of the true interferogram. Inother words, realistic dispersers will envelope or multiplythe true interferogram amplitude by a function which isbumpy and non-uniform. This is undesirable. This in-strument enveloping behavior can be determined throughcalibration procedures where a known spectrum is mea-sured. The instrumental enveloping behavior is dividedout the of the interferogram segment Q. Then to facilitategradual transition in the concatenation between neigh-boring segments, an artificially created envelope 403 hav-ing the shape of a trapezoid with sloping sides is multi-plied against segment Q. That way, in the transition re-gions where the segments overlap and sum, the concate-nated interferogram will have the proper total amplitudewhile gradually changing from one segment to another.

Acknowledgments


6-22

IX. CLAIMS

(Submitted, not official version)1. An apparatus for measuring the spectral characteristics of a

multifrequency source of electromagnetic radiation, comprising:means for receiving a beam of electromagnetic radiation from

said source; andmeans for producing a vector spectrum from said electromag-

netic radiation.2. The apparatus of claim 1, wherein said means for producing

a vector spectrum from said electromagnetic radiation comprises;means for dispersing said electromagnetic radiation into individ-

ual channels organized by wavelength;means for interfering said electromagnetic radiation with a de-

layed copy of itself to produce fringes; andfor at least one individual channel, means for determining fringe

phase and amplitude of the sinusoidal component of said fringes.3. The apparatus of claim 2, further comprising means for

dithering the amount of delay between said electromagnetic ra-diation and said delayed copy by at least 1/2 of a wavelength toseparate the sinusoidal variation due to fringes from the sinusoidalvariation due to noise.

4. The apparatus of claim 1, wherein said means for producinga vector spectrum comprises:

means for interfering said electromagnetic radiation with a de-layed copy of itself to produce an interfered beam, wherein saidinterfered beam comprises fringes;

means for dispersing said interfered beam into independent chan-nels organized by wavelength to create a fringing spectrum; and

for at least one individual channel, means for determining thefringe phase and amplitude of the sinusoidal component of saidfringing spectrum to produce a vector spectrum.

5. The apparatus of claim 1, wherein the step of producing avector spectrum comprises:

means for dispersing said electromagnetic radiation into inde-pendent channels organized by wavelength to create a fringing spec-trum;

means for interfering said electromagnetic radiation with a de-layed copy of itself to produce an interfered beam, wherein saidinterfered beam comprises fringes; and

for at least one individual channel, determining the fringe phaseand amplitude of the sinusoidal component of said fringing spec-trum to produce a vector spectrum.

6. The apparatus of claim 4, wherein said fringes comprise atleast one half wave of spatial delay change that is spatially splayedacross said interfered beam in a direction that is perpendicular tothe dispersion direction of said means for dispersing said interferedbeam.

7. The apparatus of claim 6, wherein said spatial delay changeacross said beam measured in waves is less than the relative spectralresolution of said means for dispersing said interfered beam, whichis the ratio of the wavelength divided by the blurring of the slit inthe dispersion direction (λ/∆λ).

8. The apparatus of claim 4, wherein said means for dispersingsaid interfered beam comprises a slit, wherein said spatial delaychange across said beam measured in waves is at least the rela-tive spectral resolution of said means for dispersing said interferedbeam, which is the ratio of the wavelength divided by the blurringof said slit in the dispersion direction.

9. The apparatus of claim 6, wherein said spatial delay changeacross said beam occurs in discrete steps.

10. The apparatus of claim 4, wherein said means for interfer-ing said electromagnetic radiation with a delayed copy of itself toproduce an interfered beam comprises an interferometer, whereinsaid interferometer further comprises a stepped mirror, whereinsaid discrete steps are accomplished by said stepped mirror whichdefines a path length of said beam in said interferometer, whereinsaid interferometer further comprises a stepped etalon, wherein astepped etalon may be used in conjuction with said stepped mirror.

11. The apparatus of claim 6, wherein said spatial delay changefor all positions across said beam can be incremented versus timeto produce an incremented delay change, wherein the average delayfor a given exposure of said detector can be made different thansaid average delay for a later exposure, wherein an incrementeddelay change is optimally an even fraction of a wave such as onequarter wave or one third wave, wherein this is called phase step-ping, wherein the total travel of a sequence of incremented phasechanges is optimally an integer number of waves.

12. The apparatus of claim 6, wherein said spatial delay change,for all positions across said beam, can be incremented versus time,wherein an average delay for a given exposure of a detector canbe made different than said average delay for a later exposure,wherein said increment is optimally an even fraction of a wave suchas one quarter wave or one third wave, wherein this is called phasestepping, wherein the total travel of a sequence of said incrementsis optimally an integer number of waves.

13. The apparatus of claim 6, wherein said spatial delay changevaries spatially less than half a wave across said beam, whereinsaid fringe is said to be taller than the beam and spatially unre-solved, wherein said phase and said amplitude of said fringe fora given wavelength channel of said independent channels is deter-mined using two or more phase stepping exposures and an assumedsinusoidal dependence of said fringe intensity with phase steppingphase.

14. The apparatus of claim 6, wherein said spatial delay changevaries spatially by at least one half a wave across said beam,wherein said phase and said amplitude of said fringe for a givenwavelength channel of said independent channels can be determinedfrom its spatial variation across said beam for a single exposure.

15. The apparatus of claim 14, wherein additional phase step-ping exposures improve the determination of fringe phase and am-plitude for a given wavelength channel apart from common modeerrors, by assuming said fringe phase varies sinusoidally with phasestepping phase and assuming common mode errors are stationarywith respect to phase stepping phase.

16. The apparatus of claim 4, wherein said means for dispers-ing comprises a disperser having a spectral resolution, wherein saidmeans for interfering comprises an interferometer having a spectralcomb, wherein said spectral resolution of said disperser is suffi-cient to resolve said spectral comb of said interferometer, whereinsaid interferometer has a periodicity along the dispersion axis ofλ2/(delay).

17. The apparatus of claim 4, wherein said means for dispers-ing comprises a disperser having a spectral resolution, wherein thespectral resolution of said disperser is insufficient to resolve thespectral comb of said interferometer, which has a periodicity alongthe dispersion axis of λ2/(delay).

18. The apparatus of claim 4, further including a spectral ref-erence which is recorded together with a target beam so that saidvector spectrum contains components of both target and reference.

19. The apparatus of claim 18, wherein said spectral reference isan absorption spectrum, such as provided by an iodine vapor cell.

20. The apparatus of claim 18, wherein said spectral referencehas an emission spectrum.

21. The apparatus of claim 20, wherein said emission spectrumcomprises a thorium lamp.

22. The apparatus of claim 4, further comprising at least oneadditional interferometers in series with said beam, wherein eachinterferometer of said at least one additional interferometers im-prints additional components in said vector spectrum.

23. The apparatus of claim 4, wherein said means for interfer-ing comprises an interferometer, wherein said interferometer is aMichelson interferometer, wherein an input beam is split into twopaths which are interfered to produce an output.

24. The apparatus of claim 23, wherein said Michelson is asuperimposing interferometer, wherein rays of said two paths su-perimpose in said output.

25. The apparatus of claim 4, wherein said means for interfer-ing comprises an interferometer, wherein said interferometer is a

6-23

Fabry-Perot interferometer, wherein an input beam enters into arecirculating path, which effectively interferes an infinite series ofcopies of said input beam having geometrically decreasing ampli-tudes.

26. The apparatus of claim 25, wherein Fabry-Perot com-prises partially reflective mirrors having a reflectance that producesfringes that are approximately sinusoidal.

27. The apparatus of claim 26, wherein said fringes comprise anonsinusoidal component, wherein said nonsinusoidal componentsof said fringes are discriminated against by sampling said fringesat four or less discrete places per period.

28. The apparatus of claim 4, wherein a dot product operationbetween said vector spectrum and an assumed component of saidvector spectrum yields a rotational position and magnitude of anactual vector spectrum.

29. The apparatus of claim 28, wherein said dot product oper-ation includes for each wavelength channel a dot product betweenspatial components of said vector spectrum and said vector com-ponent, to form a channelized dot product.

30. The apparatus of claim 29, wherein said dot product oper-ation includes summing or averaging said channelized dot productover groups of wavelength channels to produce a generalized dotproduct.

31. The apparatus of claim 28, wherein said vector spectrum isexpressed as a linear combination of assumed vector spectrum com-ponents, wherein rotation and magnitude of said vector spectrumcomponents are solved for by applying dot products between linearcombination and individual assumed vector spectrum components.

32. The apparatus of claim 4, wherein a Fourier transformoperation applied to said vector spectrum produces an interfero-gram segment, wherein said interferogram segment can be shiftedin delay-space by an amount equal to said interferometer delay toproduce an adjusted interferogram segment, wherein said adjustedinterferogram segment represents a measurement of a portion ofa theoretical interferogram of the vector spectrum to invert theinstrument behavior that generates Moire fringes in said vectorspectrum from said input spectrum.

33. The apparatus of claim 32, wherein said adjusted interfer-ogram segment can be concatenated with other interferogram seg-ments which have different delay values to produce a concatenatedinterferogram, wherein said concatenated interferogram representsa measurement of a theoretical interferogram of said vector spec-trum, wherein said concatenation process produces a more accuraterepresentation of said theoretical interferogram.

34. The apparatus of claim 4, wherein said means for interferingcomprises an interferometer, wherein said interferometer is formedby a long baseline interferometer, wherein light from a target iscollected at two places separated by a baseline distance, whereinchanges in angular position of said target relative to said base-line produce changes in arrival times between two said beams at abeamsplitter of said interferometer, wherein said changes in arrivaltime are equivalent to changes in the delay of said interferometer,wherein changes in angular position of a target can be inferred fromcorresponding changes in phase of said vector spectrum.

35. The apparatus of claim 34, wherein a multiplicative spec-tral reference is inserted into the optical path of said beam at aplace where it imprints a spectrum of both said beams by the samesaid spectral reference, after said beamsplitter and at said separatecollection places if two identical references are used.

36. The apparatus of claim 35, wherein said multiplicative spec-tral reference is an absorptive spectral reference, wherein said ref-erence spectrum has many narrow spectral features having stablecenter wavelengths.

37. The apparatus of claim 36, wherein said absorptive spectralreference comprises an iodine vapor cell.

38. The apparatus of claim 4, wherein the illumination from ad-ditional targets collected passed through said interferometer alonga common path with said beam produce a plurality of vector spec-trums each containing several components corresponding to eachsaid additional targets, wherein relative changes in phase of said

components represent relative changes in angular position of tar-gets, wherein this can be inferred independent of detailed knowl-edge of said optical path lengths between said collection ports whicheffect all said target light in common.

39. A method for measuring the spectral characteristics of amultifrequency source of electromagnetic radiation, comprising:

receiving a beam of electromagnetic radiation from said source;and

producing a vector spectrum from said electromagnetic radia-tion.

40. The method of claim 39, wherein said step for producing avector spectrum from said electromagnetic radiation comprises;

dispersing said electromagnetic radiation into individual chan-nels organized by wavelength;

interfering said electromagnetic radiation with a delayed copy ofitself to produce fringes; and

for at least one individual channel, determining fringe phase andamplitude of the sinusoidal component of said fringes.

41. The method of claim 40, further comprising dithering theamount of delay between said electromagnetic radiation and saiddelayed copy by at least 1/2 of a wavelength to separate the sinu-soidal variation due to fringes from the sinusoidal variation due tonoise.

42. The method of claim 39, wherein the step for producing avector spectrum comprises:

interfering said electromagnetic radiation with a delayed copy ofitself to produce an interfered beam, wherein said interfered beamcomprises fringes;

dispersing said interfered beam into independent channels orga-nized by wavelength to create a fringing spectrum; and


43. The method of claim 39, wherein the step of producing avector spectrum comprises:

dispersing said electromagnetic radiation into independent chan-nels organized by wavelength to create a fringing spectrum;

interfering said electromagnetic radiation with a delayed copy ofitself to produce an interfered beam, wherein said interfered beamcomprises fringes; and


44. The method of claim 42, wherein said fringes comprise atleast one half wave of spatial delay change that is spatially splayedacross said interfered beam in a direction that is perpendicular tothe dispersion direction of said means for dispersing said interferedbeam.

45. The method of claim 44, wherein said spatial delay changeacross said beam measured in waves is less than the relative spectralresolution of said means for dispersing said interfered beam, whichis the ratio of the wavelength divided by the blurring of the slit inthe dispersion direction (l/∆l).

46. The method of claim 44, wherein said means for dispersingsaid interfered beam comprises a slit, wherein said spatial delaychange across said beam measured in waves is at least the rela-tive spectral resolution of said means for dispersing said interferedbeam, which is the ratio of the wavelength divided by the blurringof said slit in the dispersion direction.

47. The method of claim 44, wherein said spatial delay changeacross said beam occurs in discrete steps.

48. The method of claim 47, wherein the step of interfering saidelectromagnetic radiation with a delayed copy of itself to producean interfered beam comprises an interferometer, wherein said inter-ferometer further comprises a stepped mirror, wherein said discretesteps are accomplished by said stepped mirror which defines a pathlength of said beam in said interferometer, wherein said interferom-eter further comprises a stepped etalon, wherein a stepped etalonmay be used in conjuction with said stepped mirror.

6-24

49. The method of claim 44, wherein said spatial delay changefor all positions across said beam can be incremented versus timeto produce an incremented delay change, wherein the average delayfor a given exposure of said detector can be made different thansaid average delay for a later exposure, wherein an incrementeddelay change is optimally an even fraction of a wave such as onequarter wave or one third wave, wherein this is called phase step-ping, wherein the total travel of a sequence of incremented phasechanges is optimally an integer number of waves.

50. The method of claim 42, wherein said spatial delay change,for all positions across said beam, can be incremented versus time,wherein an average delay for a given exposure of a detector canbe made different than said average delay for a later exposure,wherein said increment is optimally an even fraction of a wave suchas one quarter wave or one third wave, wherein this is called phasestepping, wherein the total travel of a sequence of said incrementsis optimally an integer number of waves.

51. The method of claim 42, wherein said spatial delay changevaries spatially less than half a wave across said beam, whereinsaid fringe is said to be taller than the beam and spatially unre-solved, wherein said phase and said amplitude of said fringe fora given wavelength channel of said independent channels is deter-mined using two or more phase stepping exposures and an assumedsinusoidal dependence of said fringe intensity with phase steppingphase.

52. The method of claim 42, wherein said spatial delay changevaries spatially by at least one half a wave across said beam,wherein said phase and said amplitude of said fringe for a givenwavelength channel of said independent channels can be determinedfrom its spatial variation across said beam for a single exposure.

53. The method of claim 52, wherein additional phase steppingexposures improve the determination of fringe phase and amplitudefor a given wavelength channel apart from common mode errors,by assuming said fringe phase varies sinusoidally with phase step-ping phase and assuming common mode errors are stationary withrespect to phase stepping phase.

54. The method of claim 42, wherein the step for dispersingcomprises a disperser having a spectral resolution, wherein saidmeans for interfering comprises an interferometer having a spectralcomb, wherein said spectral resolution of said disperser is suffi-cient to resolve said spectral comb of said interferometer, whereinsaid interferometer has a periodicity along the dispersion axis ofλ2/(delay).

55. The method of claim 42, wherein the step for interferingcomprises an interferometer having a spectral comb, wherein saidvector spectrum is numerically blurred to diminish said spectralcomb and enhance Moire fringes between said spectral comb and atarget spectrum.

56. The method of claim 42, wherein the spectral resolution ofsaid disperser is insufficient to resolve the spectral comb of saidinterferometer, which has a periodicity along the dispersion axis ofλ2/(delay).

57. The method of claim 42, wherein the sum bandwidth of saidmethod is wide enough to allow fringes on separate wavelengthchannels that differ by at least 90 degrees.

58. The method of claim 42, further including a spectral refer-ence which is recorded together with a target beam so that saidvector spectrum contains components of both target and reference.

59. The method of claim 56, wherein said spectral reference isan absorption spectrum, such as provided by an iodine vapor cell.

60. The method of claim 56, wherein said spectral reference hasan emission spectrum.

61. The method of claim 58, wherein said emission spectrumcomprises a thorium lamp.

62. The method of claim 42, further comprising at least oneadditional interferometers in series with said beam, wherein eachinterferometer of said at least one additional interferometers im-prints additional components in said vector spectrum.

63. The method of claim 42, wherein said step for interfer-ing comprises an interferometer, wherein said interferometer is a

Michelson interferometer, wherein an input beam is split into twopaths which are interfered to produce an output.

64. The method of claim 63, wherein said Michelson is a super-imposing interferometer, wherein rays of said two paths superim-pose in said output.

65. The method of claim 42, wherein said means for interfer-ing comprises an interferometer, wherein said interferometer is aFabry-Perot interferometer, wherein an input beam enters into arecirculating path, which effectively interferes an infinite series ofcopies of said input beam having geometrically decreasing ampli-tudes.

66. The method of claim 65, wherein Fabry-Perot comprises par-tially reflective mirrors having a reflectance that produces fringesthat are approximately sinusoidal.

67. The method of claim 66, wherein said fringes comprise anonsinusoidal component, wherein said nonsinusoidal componentsof said fringes are discriminated against by sampling said fringesat four or less discrete places per period.

68. The method of claim 42, wherein a dot product operationbetween said vector spectrum and an assumed component of saidvector spectrum yields a rotational position and magnitude of anactual vector spectrum.

69. The method of claim 68, wherein said dot product operationincludes for each wavelength channel a dot product between spatialcomponents

of said vector spectrum and said vector component, to form achannelized dot product.

70. The method of claim 69, wherein said dot product opera-tion includes summing or averaging said channelized dot productover groups of wavelength channels to produce a generalized dotproduct.

71. The method of claim 39, wherein said vector spectrum isexpressed as a linear combination of assumed vector spectrum com-ponents, wherein rotation and magnitude of said vector spectrumcomponents are solved for by applying dot products between linearcombination and individual assumed vector spectrum components.

72. The method of claim 42, further comprising applying aFourier transform operation to said vector spectrum to producean interferogram segment, wherein said interferogram segment canbe shifted in delay-space by an amount equal to said interferome-ter delay to produce an adjusted interferogram segment, whereinsaid adjusted interferogram segment represents a measurement ofa portion of a theoretical interferogram of the vector spectrum toinvert the instrument behavior that generates Moire fringes in saidvector spectrum from said input spectrum.

73. The method of claim 72, wherein said adjusted interfero-gram segment can be concatenated with other interferogram seg-ments which have different delay values to produce a concatenatedinterferogram, wherein said concatenated interferogram representsa measurement of a theoretical interferogram of said vector spec-trum, wherein said concatenation process produces a more accuraterepresentation of said theoretical interferogram.

74. The method of claim 42, wherein the step for interferingcomprises an interferometer, wherein said interferometer is formedby a long baseline interferometer, wherein light from a target iscollected at two places separated by a baseline distance, whereinchanges in angular position of said target relative to said base-line produce changes in arrival times between two said beams at abeamsplitter of said interferometer, wherein said changes in arrivaltime are equivalent to changes in the delay of said interferometer,wherein changes in angular position of a target can be inferred fromcorresponding changes in phase of said vector spectrum.

75. The method of claim 42, wherein a multiplicative spectralreference is inserted into the optical path of said beam at a placewhere it imprints a spectrum of both said beams by the same saidspectral reference, after said beamsplitter and at said separate col-lection places if two identical references are used.

76. The method of claim 75, wherein said multiplicative spectralreference is an absorptive spectral reference, wherein said reference

6-25

spectrum has many narrow spectral features having stable centerwavelengths.

77. The method of claim 76, wherein said absorptive spectralreference comprises an iodine vapor cell.

78. The method of claim 42, wherein the illumination from ad-ditional targets collected passed through said interferometer alonga common path with said beam produce a plurality of vector spec-trums each containing several components corresponding to eachsaid additional targets, wherein relative changes in phase of saidcomponents represent relative changes in angular position of tar-gets, wherein this can be inferred independent of detailed knowl-

edge of said optical path lengths between said collection ports whicheffect all said target light in common.

79. The apparatus of claim 4, wherein said apparatus comprisesa total bandwidth that is wide enough to allow fringes on separatewavelength channels that differ by at least 90.

80. The apparatus of claim 4, wherein said means for interferingcomprises an interferometer having a spectral comb, wherein saidvector spectrum is numerically blurred to diminish said spectralcomb and enhance Moire fringes between said spectral comb and atarget spectrum.

7-1

Part 7

Examples in the Scientific LiteratureArticles in the scientific literature demonstrating or further describing the patented technologies include the followingabstracts, with excerpted Figures.

I. WHITE LIGHT VELOCITY INTERFEROMETRY

Involves measuring the velocity of an external target by active illumination, where the illumination is from aninexpensive incoherent white light source rather than a coherent laser as in the prior art. The coherence propertiesof the illumination is altered by passing it through a preparatory interferometer. See pages 1-11, 1-15, and 1-19 of5,642,194, and 5-30 of 6,115,121.

1. D. J. Erskine & N. C. Holmes, “White Light Velocimetry”, Nature 377, 317-320 (1995).

Abstract We demonstrate with white light from an incandescent lamp a generic method for using broad-band incoherent radiation in a Doppler velocity interferometer (Fig. 1). Laser illumination is no longerrequired. The principle is applicable to any wave phenomenon for which interferometers can be con-structed, such as light, sound, and microwaves. Two interferometers are used in series, with the reflectingtarget illuminated by light passing through the first interferometer. When the interferometer delays matchwithin a coherence length of the illumination, partial fringes are produced which shift with target velocitysimilarly to previous monochromatic systems. Benefits include absolute velocity determination (lack offringe skip), unambiguous multiple velocity detection, the ability to use low cost illumination sources, in-dependence from illumination wavelength including the use of chirped lasers, and the use of pulsed lasersnormally lacking sufficient coherence.

0th fringe

a)0 m/s

0th fringe

b)16 m/s

Symmetry

Symmetry

FIG. 1: Illustration from Ref. 1. Fringe shift between (a) and (b) demonstrates measurement of 16 m/s motion of a fan bladeusing incoherent broadband light from an incandescent lamp.

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(a) Before shot (b) During shotFixedpaper

Moving graph-paper

~1/2 frngshift

~1 frngshift

FIG. 2: Illustration from Ref. 2. Color photograph of multi-color fringes produced by the white light velocimeter prior (a) andduring (b) a shot. The fringes were recorded on Kodak Royal 1000 color negative film. The illumination source was a consumergrade camera flash, of 20 µs duration. The target was a stationary piece of white paper (left side with dot) overlappingnon-uniformly moving graph paper behind it (right side with blue grid lines). The fringes on the graph paper side have shiftedvertically due to velocity. The shift varies with position (top to bottom) because of a velocity gradient.

2. D. J. Erskine & N. C. Holmes, “Imaging White Light VISAR”, in SPIE Conf. Proc. 2869, High Speed Photographyand Photonics, ed. D. Paisley, 1080-1083 (1997).

Abstract An imaging white light velocimeter consisting of two image superimposing Michelson interfer-ometers in series with the target interposed is demonstrated. Interferometrically measured 2-dimensionalvelocity maps can be made of moving surfaces using unlimited bandwidth incoherent and extended areasources. Short pulse and broadband chirped pulse lasers can be used to provide temporal resolution notpossible with monochromatic illumination. A ∼20 m/s per fringe imaging velocimeter is demonstratedusing an ordinary camera flash for illumination (Fig. 2). Radial and transverse velocity components canbe measured when the illuminating and viewing beams are non-parallel.

II. DISPERSIVE INTERFEROMETRY (ASTRONOMICAL VELOCIMETRY)

Involves measuring the Doppler velocity of an external source (such as a star) through its own emitted lightrather than active illumination (the source must have spectral features). An interferometer is added in series to thetraditional diffraction grating. The technology improves upon the prior art by allowing much smaller instrumentswhich are more robust to instrumental noise. See pages 6-1 through 6-15 of 6,351,307.

3. D. J. Erskine, “An Externally Dispersed Interferometer Prototype for Sensitive Radial Velocimetry: Theory andDemonstration on Sunlight”, Publ. Astron. Soc. Pacific. 115, 255-269 (2003).

Abstract A new analytical theory of operation of a wideband interferometric Doppler spectroscopy tech-nique, called Externally Dispersed Interferometry (EDI), is presented. The first EDI prototype was testedon sunlight and detected the 12 m/s amplitude lunar signature in Earth’s motion (Fig. 3). The hybridinstrument is an undispersed Michelson interferometer having fixed delay of about 1 cm, in series with anexternal spectrograph of about 20000 resolution. The Michelson provides the Doppler shift discrimination,while the external spectrograph boosts net white light fringe visibility by reducing crosstalk from adjacentcontinuum channels. A Moire effect between the sinusoidal interferometer transmission and the inputspectrum heterodynes high spectral details to broad Moire patterns, which carry the Doppler informationin its phase. These broad patterns survive the blurring of the spectrograph, which can have several timeslower resolution than grating-only spectrographs typically used now for the Doppler planet search. Thisenables the net instrument to be dramatically smaller in size (∼ 1 m) and cost. The EDI behavior iscompared and contrasted with a previous kind of hybrid, and to the conventional grating-only technique.The inclusion of a light efficient interferometer is shown to improve the net photon signal to noise ratio,

7-3

when the nonfringing and fringing velocity information are combined. The fringing Doppler informationalone can be competitive with or larger than the nonfringing component, depending on the spatial sizeof the source relative to the slit, and on the bandwidths utilized. The EDI is theoretically shown to bedramatically less sensitive to PSF shape and centroid variations. This could allow use of wider bandwidthspectral fiducial sources and gratings optimized for highest diffraction efficiency.

7/1 7/6 7/11 7/16 7/21 7/26Date (1998)

∆V(m/s)

40

20

0

-20

-40EDI data

Theory:

Detection of lunar signature inDoppler shift of sunlightUsing LLNL's Externally Dispersed Interferometer

12 m/s tugging of moon on Earth(Same as Jupiter on Sun)

FIG. 3: Illustration from Ref. 3. The early EDI prototype detected the 12 m/s amplitude Doppler signature of the moontugging the Earth. Sunlight fringing spectra were measured and the effects of the Earth’s orbit and daily rotation subtracted,leaving the wobble caused by the moon’s gravitational pull. This is the same size effect as a Jupiter-like planet pulling a star.The 8 m/s scatter is believed due to pointing error of the rooftop heliostat against the 4000 m/s velocity gradient of the solardisk. The estimated ±4 m/s long term instrument error is based on bromine-iodine tests under similar configuration.

III. DISPERSIVE INTERFEROMETRY (ANGULAR MEASUREMENT)

Involves spectrally dispersing the output of a long baseline interferometer, detecting a sinusoidal variation vswavelength, and assigning the periodicity to the angle of the source. Improves upon the prior art by adding significantspectral dispersion, which allows use on multiple sources and hence use in a differential mode. The latter allowsaccurate measurements in spite of drifts in the optical path length of the baseline, a major engineering problem forthe prior art. See pages 6-17 through 6-19 of 6,351,307.

4. D. J. Erskine & J. Edelstein, “Spectral Astrometry Mission for Planets Detection”, in SPIE Proc. 4852,Interferometry in Space, ed. M. Shao, 695-706 (2003).

Abstract The Spectral Astrometry Mission is a space-mission concept that uses simultaneous, multiple-star differential astrometry to measure exo-solar planet masses. The goal of SAM is to measure the reflexmotions of hundreds of nearby (∼50 pc) F, G and K stars, relative to adjacent stars, with a resolution of2.5 µ-arcsec. SAM is a new application of Spectral Interferometry (SI), also called Externally DispersedInterferometry (EDI), that can simultaneously measure the angular difference between the target andmultiple reference stars. SI has demonstrated the ability to measure a λ/20, 000 white-light fringe shiftwith only λ/3 baseline control. SAM’s structural stability and compensation requirements are thereforedramatically reduced compared to existing long-arm balanced-arm interferometric astrometry methods.We describe the SAM’s mission concept, long-baseline SI astrometry method, and technical challenges toachieving the mission.

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Beats in fringingspectra yield ∆

CCD

B

1

2

∆

1

2

a) Both sources

b) Source 1

c) Source 2

FIG. 4: Illustration from Ref. 4. Laboratory data demonstration of a multi-object long baseline spectral interferometer (SI).Two stars observed by the spectral interferometer each would produce a unique fringe pattern, which manifests beats whensummed at the detector. The white light fringe patterns shown were obtained using the prototype SI modified to have a baselineB, and two white light pinhole sources simulating stars, both together (#a with beats) and separately (#b, #c). Pinholes wereseparated by 0.5 with B=6 mm and spectrum bandwidth was 350 A over 5400 A.

IV. DISPERSIVE INTERFEROMETRY (HIGH RESOLUTION SPECTROSCOPY)

Involves inserting a small interferometer in series with a conventional spectrograph, recording the moire patternsproduced in the spectrum, and numerically processing these to recover high resolution spectral information otherwisenot detectable by the spectrograph alone. Improves upon the prior art by dramatically reducing the spectrograph sizeneeded to achieve a given resolution. Relative to Fourier Transform Spectrometers it improves the photon limitedsignal to noise ratio by 1 or 2 orders of magnitude. See pages 6-19 through 6-21 of 6,351,307.

5. D. J. Erskine, J. Edelstein, W. M. Feuerstein & B. Welsh, “High Resolution Broadband Spectroscopy using anExternally Dispersed Interferometer”, ApJ 592, L103-L106 (2003).

Abstract An externally dispersed interferometer (EDI) is a series combination of a fixed delay interfer-ometer and an external grating spectrograph. We describe how the EDI interferometer can boost theeffective resolving power of an echelle or linear grating spectrograph by a factor of 2-3 or more over thespectrograph’s full-bandwidth. The interferometer produces spectral fringes over the entire spectrographbandwidth. The fringes heterodyne with spectral features to provide a low spatial-frequency moire pat-tern. The heterodyning is numerically reversed to recover highly detailed spectral information that wouldotherwise be unattainable by the spectrograph alone. We demonstrate resolution boosting for stellar andsolar measurements of 2d echelle and linear grating spectra. Effective spectral resolution of ∼100,000 has

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b) Ordinary solar spectrum at R~20 k

"Bass"

d) Heterodyning reversed numerically

"Treble"

5140513851365134Wavelength (Å)

e) Composite "Bass" + "Treble" Actual solar spectrum at R=60 k

c) Fringing spectrum (1of 3 phases)

a) CCD data

0

Inte

nsity

(R

el.)

1

FIG. 5: Illustration from Ref. 5. Demonstration of EDI 2.5× resolution boosting for the solar spectrum using a R∼20,000linear grating spectrograph and τ= 1.1 cm interferometer delay. (a) CCD recording of an EDI fringing spectrum where φ variestransversely to dispersion (multi-phase recording). (b) The ordinary spectrum Bord(λ) derived from a histogram of (a). (c) Afringing spectrum phase exposure Bφ,from a lineout of (a) is used to obtain (d), the fringing spectrum Bedi(λ). (e) The EDIcomposite spectrum(upper curve), after equalization using iodine calibration spectra, is nearly identical to a reference KittPeak FTS solar spectrum (lower curve) artificially blurred to R=60,000. Data was taken June 16, 1998 at Lawrence LivermoreNat. Lab. Intensities are relative and have been offset.

been obtained from the ∼50,000 resolution Lick Observatory 2d echelle spectrograph, and of ∼50,000 froma ∼20,000 linear grating spectrograph (Fig. 5).

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